YOP leaders-in-training cross a glowing stretch of the New Jersey Appalachian Trail known as Raccoon Ridge. Photo by Ryan Smith.
Trainees break the ice on day 1. Photo by Ryan Smith.
Venuat demonstrates how to tie a knot. Photo by Ryan Smith.
Bernel Harrison practices knot tying as part of YOP's training for leading youth outdoors. Photo by Ryan Smith.
The YOP gear library at Mohican Outdoor Center. Photo by Ryan Smith.
Chawansky (center) and Gilson (right) consult a map en route to the group's backcountry campsite. Photo by Ryan Smith.
The group reaches its backcountry campsite on the Appalachian Trail. Photo by Ryan Smith.
Chawansky shares thoughts on his progress mid-trip. Photo by Ryan Smith.
McCrann congratulates Harrison on completing the YOP training. Photo by Ryan Smith.
Morning is still asserting itself—early light eking through dusty windows, aroma of freshly brewed coffee wafting in from the kitchen—as eight strangers settle into mismatched couches and chairs on an unseasonably warm Thursday in October.
The group gives off a mix of excitement, anxiety, and grogginess, some having driven from as far as New Hampshire to get here. "Here" is AMC's Mohican Outdoor Center, a rustic camp nestled in a grove of towering white pines and red oaks not far from the Delaware River. Although Mohican—located in the Delaware Water Gap National Recreation Area, straddling the Pennsylvania–New Jersey border (see "Best of the Delaware River,")—is open to the public, this group means business.
At the stroke of 8 a.m., Sebastien Venuat gets started. The manager of AMC's Youth Opportunities Program for the New York–New Jersey region and an instructor of the training for which this group has assembled, Venuat shares his expectations clearly but with a trademark gleam in his eye. "Starting now, everyone shall forfeit the use of cell phones and electronic devices unless it's absolutely necessary," he says, locking eyes with each participant in turn. "I want you all in for this program—no distractions." The attendees nod in unison, accepting his contract.
Along with a volunteer, Venuat is here to instruct this collection of teachers and youth workers in leading teens on backpacking trips. The workshop, known within AMC as an Outdoor Leadership Training, is key to the organization's Youth Opportunities Program (YOP), which celebrates its 50th anniversary this year. Once the attendees have successfully completed the training, they'll become YOP members, meaning they'll have access to AMC resources, from staff advice to lodging discounts at AMC destinations to free use of vast amounts of gear.
YOP has used this train-and-support model since its inception in 1968. Arising from the Civil Rights movement, the program originally focused on under-resourced youth and youth workers in Boston's Roxbury neighborhood. In its first year, AMC trained three adults who led 63 kids on outdoor trips. In the auspicious half-century since, YOP has trained more than 5,000 leaders—teachers and youth workers, like those at Mohican—who in turn have led more than 294,000 youth on outdoor trips. With support from grant funding and generous donors, YOP reached more than 37,000 youth in 2016 and aims to increase that number to 50,000 youth annually by 2020.
For some participants, the training is the first time they'll light a camp stove, sleep in a tent, or lace up a pair of hiking boots. The thought of mastering those unfamiliar tasks and then organizing and leading others—teenagers, no less—to do the same, often far from home, is daunting. While John McCrann, Venuat's volunteer co-instructor for the weekend and a high school math teacher in New York, agrees the backpacking skills are critical and complex, he believes they're secondary to even more difficult work the participants have already done on their own.
For Bernel Harrison, a police officer assigned to New Jersey's Plainfield High School, his biggest challenge isn't conquering self-doubt or finding kids to take on backpacking trips. Instead, it's a hurdle all too familiar to most of his fellow trainees: funding. Perpetual budget constraints make it tough for schools and youth groups to outfit teens with specialized—and costly—outdoor equipment. And Harrison knows what it means not to have money for gear. As a volunteer middle-school track coach, he has paid for his athletes' running shoes out of his own pocket.
That's another place YOP comes in. YOP members can borrow everything they need for their groups—from hiking boots and rain pants to stoves, sleeping bags, snowshoes, and backpacks—free. Beyond staffing two program offices (New York and a brand-new Boston facility located at a major public-transit hub), YOP maintains 10 gear rooms, known as "libraries," in four states: Massachusetts, Connecticut, New York, and New Jersey. When planning a trip, YOP members can consult with staff on choosing the right shoes and the right trails on which to wear them.
It's this support Harrison is looking for. This summer he's helping run a three-week leadership program for middle schoolers at Plainfield's Queen City Mentoring Academy, directed by police officers. "By partnering with YOP, the academy will be able to offer overnight trips to outdoor destinations, including Mohican," he says. "It's a new element the kids will be super excited about." While leadership is at the core of the academy's curriculum, Harrison, who was introduced to YOP by a teacher in his school, believes the outdoors adds a whole extra level of growth and development. "Bringing youth to an unfamiliar environment that's out of their comfort zone allows them to let their guard down and to learn more about their inner strength," he says.
Once Venuat and McCrann wrap up the morning's orientation, they move the training outdoors, into the warm autumn air. By lunch, the group has learned emergency medical protocol, how to set up a tent and a tarp, how to tie knots, and how to model proper trip behavior. Harrison is no stranger to the latter. He spends a lot of his time talking to kids, encouraging them to choose other paths than gangs. As for his knot-tying skills, Harrison admits his trucker's hitch has a ways to go. But that's why he signed up for the training: to learn by doing.
The group spends the rest of the afternoon preparing for its first night in the field, at a tent site half a mile from the YOP gear library—close enough to ease into the unfamiliar but far enough to feel remote. After learning how to pack a backpack properly, the participants don their heavy loads with a few grunts and groans. Before setting off, Venuat suggests a roll call to ensure all are accounted for. "How about this?" Harrison, known among his high schoolers as the hip-hop cop, chimes in. "Everyone counts off by saying an assigned word from this rhyme: You down with Y–O–P? Yeah, you know me." His fellow trainees love the idea and give it a try, laughing and cheering as they go. Spirits bolstered, the group—strangers this morning—heads out, bonded as if they've known each other for years.
After more instruction at camp (setting up and using a backpacking stove, keeping a safe camp kitchen, and practicing proper camp hygiene) followed by dinner, the trainees load all of their scented toiletries, dishware, and food into the campsite's steel bear box. Harrison, who has never backpacked before, admits he's a little afraid of sleeping in black bear country but knows that getting past his nerves will help him be a more confident leader. "If my kids see that I'm not afraid, then they won't be afraid," he says. By 11 p.m., the campsite is silent, save for a great horned owl hooting from a distance in the starlit night.
Gilson echoes Chawansky, adding, "I not only wanted to learn how to stay alive on my trips but how to have a ton of fun too." Fun is on the agenda, but more immediate is a test, of sorts. Starting this morning, Venuat and McCrann will designate a series of temporary leaders, giving everyone a chance to demonstrate they have what it takes to guide a group outdoors. Gilson and Chawansky are up first. It's a good transition into the training's next phase: a three-day, 13-mile hike along Kittatinny Ridge, finishing back at the YOP gear library via the Appalachian Trail.
Eager to showcase their newfound prowess, Gilson and Chawansky oversee a speedy departure from camp and lead the group back to the parking lot, where they'll catch a shuttle to the Delaware Water Gap trailhead. During the half-hour drive, the trainees brainstorm ideas for their first YOP-supported outings. Harrison plans to host a camping-themed workshop at the local library. Chawansky wants to take his seventh-graders camping at Brooklyn's Floyd Bennett Field, in preparation for longer trips to Mohican. Gilson hopes to lead her students on an urban backpacking trip through New York City. McCrann advises the group to start teaching skills in their schools, so the kids can get comfortable with the basics in a familiar setting.
"Remember: An outdoor trip should start at your youth's comfort level, not your own," he says. He refrains from reminding them that, just yesterday, their own comfort levels were a lot lower.
Upon reaching the trailhead, Gilson and Chawansky seamlessly resume their leadership roles, reviewing a topographic map of the area with the group to take stock of the route ahead. Chawansky assists Harrison in locating water sources on the map and deciphering how much elevation they'll gain. As Gilson and Chawansky lead roll call then lift their packs and amble up the trail, Venuat and McCrann linger behind, marveling at how thoroughly the group members have internalized what they've learned.
Fast forward three days, 13 miles, two nights at remote campsites, one deluge, and countless pesky bugs, and the group returns to Mohican. Blistered, sore, and elated, they've learned a surprising amount about their capabilities as individuals and as leaders. Earlier this morning, Venuat and McCrann met with each participant individually to evaluate his or her progress. If the instructors deemed anyone in need of more experience before leading kids outdoors, they would chart a plan for further YOP training. But everyone in this group received an enthusiastic thumbs-up.
It's this self-confidence—rooted in technical mastery, good decision-making, and a passion for the outdoors—that YOP hopes to build in both youth workers and their youth. The outdoors has the power to transform a teen's world: at home, at school, and in the community. "Going outside gives kids time to reevaluate what matters in life, to listen to what's in their hearts, and to simply have the space and time to be," Chawansky says.
1968: AMC launches first training for three youth agencies from Roxbury, Mass.
Hear from participating kids firsthand in a new YOP video.
Learn more about the Youth Opportunities Program.
And support AMC's work in getting kids outside.


text:
\section{Introduction}

Let$\ (M,g)$ be a Riemannian compact smooth $n-$manifold, $n\geq 5$, with
metric $g$ and scalar curvature $S_{g}$, we let $H_{2}^{2}(M)$ be the
standard Sobolev space which is the completion of the space
\begin{equation*}
C_{2}^{2}(M)=\left\{ u\in C^{\infty }(M)\text{: }\left\Vert u\right\Vert
_{2,2}<+\infty \right\}
\end{equation*
with respect to the norm $\left\Vert u\right\Vert
_{2,2}=\sum_{l=0}^{2}\left\Vert \nabla ^{l}u\right\Vert _{2}$.

In this paper, we investigate solutions of a class of fourth order elliptic
equations, on compact $n$-dimensional Riemannian manifolds, of the form
\begin{equation}
\Delta ^{2}u+\nabla ^{i}(a(x)\nabla _{i}u)+b(x)u=f(x)\left\vert u\right\vert
^{N-2}u+\lambda \left\vert u\right\vert ^{q-2}u  \tag{1}  \label{1}
\end{equation
where $a$, $b$, and $f$ \ are smooth functions on $M$ , $N=\frac{2n}{n-4}$
is the critical exponent, $1<q<2$ \ a real number, $\lambda >0$ a real
parameter.

Consideration for such problem comes from conformal geometry: indeed, in
1983, Paneitz \cite{8} introduced a conformal fourth order operator defined
on $4$-dimensional Riemannian manifolds which was generalized by Branson
\cite{3} to higher dimensions.
\begin{equation*}
PB_{g}(u)=\Delta ^{2}u+div(-\frac{(n-2)^{2}+4}{2(n-1)(n-2)}S_{g}.g+\frac{4}
n-2}Ric)du+\frac{n-4}{2}Q^{n}u
\end{equation*
where $\Delta u=-div\left( \nabla u\right) $, $R$ is the scalar curvature,
Ric$ is the Ricci curvature of $g$ and wher
\begin{equation*}
Q^{n}=\frac{1}{2(n-1)}\Delta S_{g}+\frac{n^{3}-4n^{2}+16n-16}
8(n-1)^{2}(n-2)^{2}}S_{g}^{2}-\frac{2}{(n-2)^{2}}\left\vert Ric\right\vert
^{2}
\end{equation*
is associated to the notion of $Q$ -curvature.

We refer to a Paneitz-Branson type operator as an operator of the for
\begin{equation*}
P_{g}u=\Delta ^{2}u+\nabla ^{i}(a(x)\nabla _{i}u)+h(x)u\text{.}
\end{equation*}

Equation (\ref{1}) is a perturbation of the equation
\begin{equation}
\Delta ^{2}u+\nabla ^{i}(a(x)\nabla _{i}u)+h(x)u=f(x)\left\vert u\right\vert
^{N-2}u\text{.}  \tag{2}  \label{2}
\end{equation
Since the embedding $H_{2}^{2}\hookrightarrow H_{N}^{k}$ , ($\ k=0$, $1$)
fails to be compact, as known, one encounters serious difficulties in
solving equations like (\ref{1}).

Since 1990 many results have been established for precise functions $a$, $h$
and $f.$ D.E. Edmunds, D. Fortunato, E. Jannelli$\left( \text{\cite{8}
\right) $ proved for $n\geq 8$ that if $\lambda \in (0,\lambda _{1})$, with
\lambda _{1}$ is the first eigenvalue of $\Delta ^{2}$ on the euclidean open
ball $B$, the problem
\begin{equation*}
\left\{
\begin{array}{c}
\Delta ^{2}u-\lambda u=u\left\vert u\right\vert ^{\frac{8}{n-4}}\text{ in }B
\\
u=\frac{\partial u}{\partial n}=0\text{ on }\partial
\end{array
\right.
\end{equation*
has a non trivial solution.

In 1995, R$.$ Van der Vorst (\cite{8}) obtained the same results as D.E.
Edmunds, D. Fortunato, E. Jannelli. when applied to the proble
\begin{equation*}
\left\{
\begin{array}{c}
\Delta ^{2}u-\lambda u=u\left\vert u\right\vert ^{\frac{8}{n-4}}\text{ in
\Omega \\
u=\Delta u=0\text{ \ on \ }\partial \Omeg
\end{array
\right.
\end{equation*
where $\Omega $ is an open bounded set of $R^{n}$ and moreover he showed
that the solution is positive

In $\left( \text{\cite{7}}\right) $ D.Caraffa studied the equatio
\begin{equation}
\Delta ^{2}u+\nabla ^{i}(a(x)\nabla _{i}u)+b(x)u=f(x)\left\vert u\right\vert
^{N-2}u  \tag{3}  \label{3}
\end{equation
in the case $f(x)=$constant; and in the particular case where the functions
a(x)$ and $b(x)$ are precise constants she obtained the existence of
positive regular solutions.

Let $f$ \ be a $C^{\infty }$ function on $\ M$, $f^{-}=-\inf (f,0)$,
f^{+}=\sup (f,0)$ and for $a$,$b$, $\ f$ , $C^{\infty }$ -functions on $M$,
we let $\ \
\begin{equation*}
\lambda _{a,f}=\underset{u\in A}{\inf }\frac{\int_{M}(\Delta
u)^{2}dv_{g}-\int_{M}a\left\vert \nabla u\right\vert ^{2}dv_{g}}
\int_{M}u^{2}dvg}
\end{equation*
where $A=\left\{ u\in H_{2}\text{, }u\geq 0\text{, }u\not\equiv 0\text{ s.
t. }\int_{M}f^{-}udv_{g}=0\right\} ,~\ $an
\begin{equation*}
\lambda _{a,f}=+\infty \ \ \ \ \text{if\ \ \ \ \ \ \ \ \ }A=\phi \text{ }
\end{equation*
the second author establish the following:

\begin{theorem}
\cite{3} Let $a$, $h$ be $C^{\infty }$ functions on $M$ with $h$ negative.
For every $C^{\infty }$ function, $f$ on $M$ with $\int_{M}f^{-}dv_{g}>0$,
there exists a constant $C>0$ which depends only on $\frac{f^{-}}{\int
f^{-}dv_{g}}$ such that if $f$ satisfies the following conditions
\begin{equation*}
\left.
\begin{array}{l}
(1)\text{ }\left\vert h(x)\right\vert <\lambda _{a,f}\text{ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ for any }x\in M\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
(2)\text{ }\frac{\sup f^{+}}{\int f^{-}dv_{g}}<C \\
(3)\text{ }\sup_{M}f>0
\end{array
\right.
\end{equation*
then the subcritical equation
\begin{equation*}
\Delta ^{2}u+\nabla ^{i}(a\nabla _{i}u)+hu=f\left\vert u\right\vert ^{q-2}u
\text{ \ \ \ \ }q\in \left] 2,N\right[
\end{equation*
has at least two distinct solutions $u$ and $v$ satisfying $F_{q}\left(
u\right) <0<F_{q}\left( v\right) $ and of class $C^{4,\alpha }$, for some
\alpha \in (0,1)$.
\end{theorem}

\begin{theorem}
\cite{3} Let $a$, $h$ be $C^{\infty }$ functions on $M$ with $h$ negative.\
For every $C^{\infty }$ function $f$ on $M$ with $\int_{M}f^{-}dv_{g}>0$
there exists a constant $C>0$ which depends only on $\frac{f^{-}}{\int
f^{-}dv_{g}}$ such that if $f$ satisfies the following conditions
\begin{equation*}
\left.
\begin{array}{l}
(1)\text{ }\left\vert h(x)\right\vert <\lambda _{a,f}\text{ \ \ \ \ \ \ \
for any }x\in M\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
(2)\text{ }\frac{\sup f^{+}}{\int f^{-}dv_{g}}<
\end{array
\right.
\end{equation*
the critical equatio
\begin{equation*}
\Delta ^{2}u+\nabla ^{i}(a\nabla _{i}u)+hu=f\left\vert u\right\vert ^{N-2}u
\end{equation*
has a solution of class $C^{4,\theta }$, for some $\theta \in (0,1)$, \ with
negative energy.
\end{theorem}

Let
\begin{equation}
\Delta ^{2}u+\nabla ^{i}(a(x)\nabla _{i}u)+h(x)u=f(x)\left\vert u\right\vert
^{N-2}u+\lambda \left\vert u\right\vert ^{q-2}u+\epsilon g(x)  \tag{4}
\label{4}
\end{equation
where $a$, $h$, $f$ and $g$ are smooth functions on $M$ , $N=\frac{2n}{n-4}$
is the critical exponent, $2<q<N$ \ a real number, $\lambda >0$ a real
parameter and $\epsilon >0$ any small real number the second author obtain

\begin{theorem}
\cite{4} Let $(M,g)$ be a compact Riemannian $n-$manifold, $n\geq 6$, $a$,
h $, $f$, $g$ be smooth real \ functions on $M$ with

(i) $f(x)>0$ and $g(x)>0$ everywhere on $M$

(ii) the operator $P(u)=\Delta ^{2}u+\nabla ^{i}(a(x)\nabla _{i}u)+h(x)u$ is
coercive

(iii) if $n>6$ , we suppose $\frac{\Delta f(x_{o})}{2f(x_{o})
+C_{1}(n)S_{g}(x_{o})+C_{2}(n)a(x_{o})>0$ and if $n=6$, we suppose that
\frac{4}{3n}S_{g}(x_{o})+\frac{1}{(n-4)}a(x_{o})>0$, where $C_{1}(n)=\frac
5n^{2}(n-7)+52(n-1)}{6n(n+2)(5n-6)}$ and $\ C_{2}(n)=\frac{8(n-1)}{(n+2)(n-6
}$

Then equation (\ref{4}) has at least two distinct solutions in $H_{2}$
\end{theorem}

Our main results in this paper state as follows

\begin{theorem}
\label{th1} Let $\left( M,g\right) $ be an $n$-dimensional compact
Riemannian manifold with $n\geqslant 6$ and $f$ a positive smooth function
on $M$. Assume that the operator $P(u)=\Delta ^{2}u+\nabla ^{i}(a(x)\nabla
_{i}u)+b(x)u$ is coercive.

If $n>6$ and at the point $x_{o}$ where the function $f$ achieves its
maximum the following condition is satisfied
\begin{equation*}
\frac{n\left( n^{2}+4n-20\right) }{2\left( n-2\right) \left( n^{2}-4\right)
S_{g}\left( x_{o}\right) +\frac{n\left( n-6\right) }{\left( n-2\right)
\left( n^{2}-4\right) }a\left( x_{o}\right) -\frac{n}{8\left( n-2\right)
\frac{\Delta f(x_{o})}{f\left( x_{o}\right) }>0
\end{equation*
then there exists a $\Lambda >0$ such that for any $\lambda \in \left(
0,\Lambda \right) $, equation (\ref{1}) have a non trivial solution of class
$C^{4,\theta }(M)$, $\alpha \in (0,1)$.

If $n=6$ and the condition $S_{g}(x_{o})>-3a(x_{o})$ is satisfied, then
equation (\ref{1}) has a solution $u$ of class $C^{4,\theta }(M)$ provided
that $\lambda \in \left( 0,\Lambda \right) $.
\end{theorem}

In case of constant coefficients, equation (\ref{1}) reduces to
\begin{equation}
\Delta ^{2}u+\alpha \Delta u+\beta u=f(x)\left\vert u\right\vert
^{N-2}u+\lambda \left\vert u\right\vert ^{q-2}u  \tag{5}  \label{5}
\end{equation
where $\alpha $ and $\beta $ are real constants; we get the existence of
tree solutions.

\begin{theorem}
\label{th2} Suppose all the conditions of Theorem\ref{th1} are satisfied and
moreover $\frac{\alpha ^{2}}{4}>\beta $ with $\alpha >0$. Then equation (\re
{5}) has beside a positive $u^{+}$ and a negative $u^{-}$ smooth solutions a
third solution $w$ distinct from $u^{+}$ and $u^{-}$.
\end{theorem}

The technique relies on critical points theory. To find solutions we use a
method developed in \cite{1}. The solutions appeared as critical points
restricted to a suitable manifold. In the case of constant coefficients we
obtain the existence of two s

Consider the functionals $J_{\lambda }$, $J_{\lambda }^{+}$ and $J_{\lambda
}^{-}$ defined on $H_{2}$ by
\begin{equation}
J_{\lambda }(u)=\frac{1}{2}\left( \left\Vert \Delta u\right\Vert
_{2}^{2}-\int_{M}a(x)\left\vert \nabla u\right\vert
^{2}dv_{g}+\int_{M}b(x)u^{2}dv_{g}\right) -\frac{\lambda }{q
\int_{M}\left\vert u\right\vert ^{q}dv_{g}\text{\ \ \ }  \tag{6}  \label{6}
\end{equation
\begin{equation*}
-\frac{1}{N}\int_{M}f(x)\left\vert u\right\vert ^{N}dv_{g},
\end{equation*}
\begin{equation}
J_{\lambda }^{+}(u)=\frac{1}{2}\left( \left\Vert \Delta u\right\Vert
_{2}^{2}-\int_{M}a(x)\left\vert \nabla u\right\vert
^{2}dv_{g}+\int_{M}b(x)u^{2}dv_{g}\right) -\frac{\lambda }{q}\int_{M}\left(
u^{+}\right) ^{q}dv_{g}\text{\ \ \ }  \tag{6}
\end{equation
\begin{equation*}
-\frac{1}{N}\int_{M}f(x)\left( u^{+}\right) ^{N}dv_{g}
\end{equation*
and
\begin{equation}
J_{\lambda }^{-}(u)=\frac{1}{2}\left( \left\Vert \Delta u\right\Vert
_{2}^{2}-\int_{M}a(x)\left\vert \nabla u\right\vert
^{2}dv_{g}+\int_{M}b(x)u^{2}dv_{g}\right) -\frac{\lambda }{q
\int_{M}\left\vert u^{-}\right\vert ^{q}dv_{g}\text{\ \ \ }  \tag{6}
\end{equation
\begin{equation*}
-\frac{1}{N}\int_{M}f(x)\left\vert u^{-}\right\vert ^{N}dv_{g}
\end{equation*
where $u^{+}=\max \left( u,0\right) $ and $u^{-}=\min \left( u,0\right) $.

Let

\begin{equation*}
Q_{\lambda }\left( u\right) =\left\langle \nabla J_{\lambda }\left( u\right)
,u\right\rangle
\end{equation*}

and

\begin{equation*}
Q_{\lambda }^{\pm }\left( u\right) =\left\langle \nabla J_{\lambda }^{\pm
}\left( u\right) ,u\right\rangle
\end{equation*
where $\left\langle \nabla J_{\lambda }\left( u\right) ,v\right\rangle $
denotes the value of $\nabla J_{\lambda }\left( u\right) $ at $v$.

We consider also the se
\begin{equation*}
M_{\lambda }=\left\{ u\in H_{2}:Q_{\lambda }\left( u\right) =0\text{ and
\left\Vert u\right\Vert \geq \rho >0\right\}
\end{equation*
an
\begin{equation*}
M_{\lambda }^{\pm }=\left\{ u\in H_{2}:Q_{\lambda }^{\pm }\left( u\right) =
\text{ and }\left\Vert u\right\Vert \geq \rho >0\right\} \text{.}
\end{equation*}
Along this paper the functions $a$ and $b$ are taken such that

\begin{equation*}
\left\Vert u\right\Vert ^{2}=\left\Vert \Delta u\right\Vert
_{2}^{2}-\int_{M}a(x)\left\vert \nabla u\right\vert
^{2}dv_{g}+\int_{M}b(x)u^{2}dv_{g}
\end{equation*
is a norm on $H_{2}^{2}(M)$ equivalent to the usual one: for example by
letting $\max_{x\in M}a(x)<0$ and $\min_{x\in M}b(x)>0$ which is equivalent
to assume that the operator $P(u)=\Delta ^{2}u+a(x)\Delta u+b(x)u$ is
coercive.

First we ensure that the manifold $M_{\lambda }$ ( respectively $M_{\lambda
}^{\pm }$ ) is not empty.

\begin{lemma}
\label{lem1} There is a real $\lambda _{o}>0$ such that the set $M_{\lambda
} $ is non empty for any $\lambda \in \left( 0,\lambda _{o}\right) $.
\end{lemma}

\begin{proof}
For $u\in H_{2}^{2}\left( M\right) $ with $\left\Vert u\right\Vert
_{2,2}\geq \rho >0$ and $t>0$,
\begin{equation*}
Q_{\lambda }\left( tu\right) =t^{2}\left\Vert u\right\Vert ^{2}-\lambda
t^{q}\left\Vert u\right\Vert _{q}^{q}-t^{N}\int_{M}f(x)\left\vert
u\right\vert ^{N}dv_{g}\text{.}
\end{equation*
Put
\begin{equation*}
\alpha \left( t\right) =\left\Vert u\right\Vert
^{2}-t^{N-2}\int_{M}f(x)\left\vert u\right\vert ^{N}dv_{g}
\end{equation*
and
\begin{equation*}
\beta \left( t\right) =\lambda t^{q-2}\left\Vert u\right\Vert _{q}^{q}\text{
}
\end{equation*
The Sobolev inequality leads t
\begin{equation*}
\alpha \left( t\right) \geq \left\Vert u\right\Vert ^{2}-\max_{x\in
M}f\left( x\right) \left( \max \left( \left( 1+\epsilon \right)
K_{o},A_{\epsilon }\right) \right) ^{N}\left\Vert u\right\Vert
_{H_{2}^{2}\left( M\right) }^{N}t^{N-2}
\end{equation*
where
\begin{equation*}
\frac{1}{K_{\circ }}=\inf {}_{u\in H_{2}^{2}
\mathbb{R}
^{n})-\{0\}}\frac{\left\Vert \Delta u\right\Vert _{2}^{2}}{\left\Vert
u\right\Vert _{N}^{2}}
\end{equation*}
is the best constant in the Sobolev's embedding $H_{2}^{2}(R^{n})\subset
L^{N}(R^{n})$ (see T. Aubin \cite{2} ) and $A_{\epsilon }$ is a positive
constant depending on $\epsilon $, and since the norms $\left\Vert
.\right\Vert $ and $\left\Vert .\right\Vert _{H_{2}^{2}\left( M\right) }$
are equivalent, there is $\Lambda >0$ such that
\begin{equation}
\alpha \left( t\right) \geq \left\Vert u\right\Vert ^{2}-\Lambda ^{-\frac{N}
2}}\max_{x\in M}f\left( x\right) \left( \max \left( \left( 1+\epsilon
\right) K_{o},A_{\epsilon }\right) \right) ^{N}\left\Vert u\right\Vert
^{N}t^{N-2}\text{.}  \tag{7}  \label{7}
\end{equation
Combining the H\"{o}lder and the Sobolev inequalities and taking account of
the equivalence of the norms $\left\Vert .\right\Vert $ and $\left\Vert
.\right\Vert _{H_{2}^{2}\left( M\right) }$, we ge
\begin{equation}
\beta \left( t\right) \leq \lambda V\left( M\right) ^{1-\frac{2}{N}}\left(
\max \left( \left( 1+\epsilon \right) K_{o},A\right) \right) ^{\frac{n}{2
}\left\Vert u\right\Vert ^{q}t^{q-2}\text{.}  \tag{8}  \label{8}
\end{equation
Let $\alpha _{1}(t)$ and $\beta _{1}(t)$ denote respectively right hand
sides of inequalities (\ref{7}) and (\ref{8}); $\alpha _{1}\left( t\right) $
vanishes fo
\begin{equation}
t_{o}=\frac{1}{\left\Vert u\right\Vert \left( \max_{x\in M}f(x)\right) ^
\frac{1}{N-2}}\left( \max \left( \left( 1+\epsilon \right) K_{o},A_{\epsilon
}\right) \right) ^{\frac{N}{2\left( N-2\right) }}}\text{.}  \tag{9}
\label{9}
\end{equation
So if we choose
\begin{equation*}
\left\Vert u\right\Vert =\frac{1}{\left( \max_{x\in M}f(x)\right) ^{\frac{1}
N-2}}\left( \max \left( \left( 1+\epsilon \right) K_{o},A_{\epsilon }\right)
\right) ^{\frac{N}{2\left( N-2\right) }}}
\end{equation*
$t_{o}=1$.

Now since $\alpha _{1}(t)$ and $\beta _{1}\left( t\right) $ are both
decreasing functions, we get
\begin{equation*}
\min_{x\in \left( 0,\frac{1}{2}\right) }\alpha _{1}\left( t\right) =\alpha
_{1}(\frac{1}{2})=\frac{\left( 1-2^{2-N}\right) }{\left( \max_{x\in
M}f(x)\right) ^{\frac{2}{N-2}}\left( \max \left( \left( 1+\epsilon \right)
K_{o},A_{\epsilon }\right) \right) ^{\frac{N}{\left( N-2\right) }}}
\end{equation*
\begin{equation*}
\geq \left( 1-2^{2-N}\right) \rho ^{2}>0
\end{equation*
and
\begin{equation*}
\min_{x\in \left( 0,\frac{t_{o}}{2}\right) }\beta _{1}\left( t\right) =\beta
_{1}(\frac{1}{2})=\frac{2^{2-q}\lambda V(M)^{\left( 1-\frac{2}{N}\right) }}
\left( \max_{x\in M}f(x)\right) ^{\frac{q}{N-2}}\left( \max \left( \left(
1+\epsilon \right) K_{o},A_{\epsilon }\right) \right) ^{\frac{q-2+N}{N-2}}}>
\text{.}
\end{equation*
The equation $Q_{\lambda }(u)=0$ admits a solution if $\min_{x\in \left( 0
\frac{1}{2}\right) }\alpha _{1}\left( t\right) \geq \min_{x\in \left( 0
\frac{1}{2}\right) }\beta _{1}\left( t\right) $ that is to say if
\begin{equation*}
0<\lambda <\lambda _{o}=\frac{\left( 2^{q-2}-2^{q-N}\right) V(M)^{\left( 1
\frac{2}{N}\right) }}{\left( \max_{x\in M}f(x)\right) ^{\frac{q-2}{N-2
}\left( \max \left( \left( 1+\epsilon \right) K_{o},A_{\epsilon }\right)
\right) ^{\frac{q-2}{N-2}}}\text{.}
\end{equation*
Hence the set $M_{\lambda }$ is nonempty for any $\lambda \in \left(
0,\lambda _{o}\right) $.

Since $Q_{\lambda }^{\pm }(u)\geqslant Q_{\lambda }(u)$ the same
calculations lead to the conclusion.
\end{proof}

\section{Palais-Smale conditions}

\begin{lemma}
\label{lem2} There is $A>0$ such that $J_{\lambda }(u)\geq A>0$ ( resp.
J_{\lambda }(u)\geq A>0$ ) for any $u\in M_{\lambda }$ with $\lambda \in
\left( 0,\min \left( \lambda _{o},\lambda _{1}\right) \right) $ where
\lambda _{1}=\frac{\frac{(N-2)\Lambda ^{-\frac{q}{2}}}{2\left( N-q\right) }}
V(M)^{1-\frac{2}{N}}\left( \max \left( \left( 1+\epsilon \right)
K_{o},A_{\epsilon }\right) \right) ^{\frac{q}{2}}\rho ^{q-2}}$and $\lambda
_{o}$ is as in Lemma \ref{1}.
\end{lemma}

\begin{proof}
Let $u\in M_{\lambda }$, then
\begin{equation*}
\left\Vert u\right\Vert ^{2}=\lambda \int_{M}\left\vert u\right\vert
^{q}dv_{g}+\int_{M}f(x)\left\vert u\right\vert ^{N}dv_{g}
\end{equation*
so on $M_{\lambda }$ the functional $J_{\lambda }$ writes

\begin{equation*}
J_{\lambda }\left( u\right) =\frac{N-2}{2N}\left\Vert u\right\Vert
^{2}-\lambda \frac{N-q}{Nq}\int_{M}\left\vert u\right\vert ^{q}dv_{g}\text{.}
\end{equation*
Combining H\"{o}lder and Sobolev inequalities we obtai
\begin{equation*}
J_{\lambda }\left( u\right) \geq \frac{N-2}{2N}\left\Vert u\right\Vert
^{2}-\lambda \frac{N-q}{Nq}V(M)^{1-\frac{2}{N}}\left( \max \left( \left(
1+\epsilon \right) K_{o},A_{\epsilon }\right) \right) ^{\frac{q}{2
}\left\Vert u\right\Vert _{H_{2}^{2}\left( M\right) }^{q}
\end{equation*
and still taking account of the equivalence of the norms $\left\Vert
.\right\Vert _{H_{2}^{2}\left( M\right) }$ and $\left\Vert .\right\Vert $,
we obtai
\begin{equation*}
J_{\lambda }\left( u\right) \geq \left( \frac{N-2}{2N}-\lambda \frac{N-q}{Nq
\Lambda ^{-\frac{q}{2}}V(M)^{1-\frac{2}{N}}\left( \max \left( \left(
1+\epsilon \right) K_{o},A_{\epsilon }\right) \right) ^{\frac{q}{2}}\rho
^{q-2}\right) \left\Vert u\right\Vert ^{2}
\end{equation*
where $\Lambda >0$ is a constant.

Hence if
\begin{equation*}
0<\lambda <\frac{\frac{N-2}{2\left( N-q\right) }\Lambda ^{-\frac{q}{2}}}
V(M)^{1-\frac{2}{N}}\left( \max \left( \left( 1+\epsilon \right)
K_{o},A_{\epsilon }\right) \right) ^{\frac{q}{2}}\rho ^{q-2}}
\end{equation*
the
\begin{equation*}
J_{\lambda }\left( u\right) >0\text{ }
\end{equation*
for any $u\in M_{\lambda }$.

Since $J_{\lambda }^{\pm }\left( u\right) \geqslant J_{\lambda }\left(
u\right) $ we have the conclusion to $J_{\lambda }^{\pm }\left( u\right) $.
\end{proof}

\begin{lemma}
\label{lem3} Let $\left( M,g\right) $ be a $n$- Riemannian manifold, $n\geq
5 $. \ The following assertions are true

(i) $\left( \nabla Q_{\lambda }(u),u\right) \leq -A<0$ ( resp. $\left(
\nabla Q_{\lambda }^{\pm }(u),u\right) \leq -A<0$ ) , for $u\in M_{\lambda }$
and any $\lambda \in \left( 0,\min \left( \lambda _{o},\lambda _{1}\right)
\right) $

(ii) The critical points of $J_{\lambda }$ ( resp. $J_{\lambda }^{\pm }$ )
are points of $M_{\lambda }$ ( resp. $M_{\lambda }^{\pm }$ ).
\end{lemma}

\begin{proof}
(i) Let $u\in M_{\lambda }$, the
\begin{equation*}
\left\Vert u\right\Vert ^{2}=\lambda \int_{M}\left\vert u\right\vert
^{q}dv_{g}+\int_{M}f(x)\left\vert u\right\vert ^{N}dv_{g}
\end{equation*
and
\begin{equation*}
\left\langle \nabla Q_{\lambda }(u),u\right\rangle =2\left\Vert u\right\Vert
^{2}-\lambda q\int_{M}\left\vert u\right\vert
^{q}dv_{g}-N\int_{M}f(x)\left\vert u\right\vert ^{N}dv_{g}
\end{equation*
\begin{equation*}
=2\left\Vert u\right\Vert ^{2}-\lambda q\int_{M}\left\vert u\right\vert
^{q}dv_{g}-N\left( \left\Vert u\right\Vert ^{2}-\lambda \int_{M}\left\vert
u\right\vert ^{q}dv_{g}\right)
\end{equation*
\begin{equation*}
=\left( 2-N\right) \left\Vert u\right\Vert ^{2}+\lambda \left( N-q\right)
\left\Vert u\right\Vert _{q}^{q}\text{.}
\end{equation*}

The combination of H\"{o}lder and Sobolev inequalities allows us to writ
\begin{equation*}
\left\langle \nabla Q_{\lambda }(u),u\right\rangle \leq \left( 2-N\right)
\left\Vert u\right\Vert ^{2}+\lambda \left( N-q\right) V(M)^{1-\frac{2}{N
}\left( \max \left( 1+\epsilon \right) K_{o},A_{\epsilon }\right) ^{\frac{q}
2}}\left\Vert u\right\Vert _{H_{2}^{2}\left( M\right) }^{q}
\end{equation*
and since the norms $\left\Vert .\right\Vert $ and $\left\Vert .\right\Vert
_{H_{2}^{2}\left( M\right) }$ are equivalent, we ge
\begin{equation*}
\left\langle \nabla Q_{\lambda }(u),u\right\rangle \leq \left( \left(
2-N\right) +\lambda \left( N-q\right) V(M)^{1-\frac{2}{N}}\Lambda
^{q-2}\left( \max \left( 1+\epsilon \right) K_{o},A_{\epsilon }\right) ^
\frac{q}{2}}\rho ^{q-2}\right) \left\Vert u\right\Vert ^{2}\text{.}
\end{equation*
Hence if
\begin{equation*}
0<\lambda <\lambda _{1}=\frac{\frac{N-2}{2\left( N-q\right) }\Lambda ^{
\frac{q}{2}}}{V(M)^{1-\frac{2}{N}}\left( \max \left( 1+\epsilon \right)
K_{o},A_{\epsilon }\right) ^{\frac{q}{2}}\rho ^{q-2}}
\end{equation*
then for any $u\in M_{\lambda }$.
\begin{equation*}
\left\langle \nabla Q_{\lambda }(u),u\right\rangle <0
\end{equation*}

(ii) \ By the Lagrange multiplicators theorem we get the existence of a real
number $\mu $ such that for any $u\in M_{\lambda }$
\begin{equation*}
\nabla J_{\lambda }(u)=\mu \nabla Q_{\lambda }(u)
\end{equation*
and by testing at the point $u\in M_{\lambda }$, we obtai
\begin{equation*}
Q_{\lambda }(u)=\left\langle \nabla J_{\lambda }(u),u\right\rangle =\mu
\left\langle \nabla Q_{\lambda }(u),u\right\rangle
\end{equation*
and since $\left\langle \nabla Q_{\lambda }(u),u\right\rangle <0$, we get
necessarily that $\mu =0$;

Hence for any $u\in M_{\lambda }
\begin{equation*}
\nabla J_{\lambda }(u)=0\text{.}
\end{equation*
The same computations are carried to conclude for $\left\langle \nabla
Q_{\lambda }^{\pm }(u),u\right\rangle $ and $J_{\lambda }^{\pm }$.
\end{proof}

\begin{lemma}
\label{lem4} Let ($u_{n}$)$_{n}$ be a sequence in $M_{\lambda }$ ( resp.
M_{\lambda }^{\pm }$ ) such that

\begin{equation*}
J_{\lambda }(u_{n})\leq c\text{ ( resp. }J_{\lambda }^{\pm }(u_{n})\leq
\text{ )}
\end{equation*
and
\begin{equation*}
\nabla J_{\lambda }(u_{n})-\mu _{n}\nabla Q_{n}(u_{n})\rightarrow 0\text{ (
resp. }\nabla J_{\lambda }^{\pm }(u_{n})-\mu _{n}\nabla Q_{n}^{\pm
}(u_{n})\rightarrow 0\text{ ).}
\end{equation*}

Suppose that
\begin{equation*}
c<\frac{2}{nK_{o}^{\frac{n}{4}}\max_{x\in M}f(x)^{\frac{n}{4}-1}}
\end{equation*
then there is a subsequence of ($u_{n}$)$_{n}$ converging strongly in
H_{2}^{2}(M)$.
\end{lemma}

\begin{proof}
Let $\left( u_{n}\right) _{n}\subset M_{\lambda }
\begin{equation*}
J_{\lambda }(u_{n})=\frac{N-2}{2N}\left\Vert u_{n}\right\Vert ^{2}-\lambda
\frac{N-q}{Nq}\int_{M}\left\vert u_{n}\right\vert ^{q}dv(g)
\end{equation*}
We have \
\begin{equation*}
J_{\lambda }(u_{n})\geq \frac{N-2}{2N}\left\Vert u_{n}\right\Vert
^{2}-\lambda \frac{N-q}{Nq}\Lambda ^{-\frac{q}{2}}V(M)^{1-\frac{q}{N}}(\max
((1+\varepsilon )K_{\circ },A_{\varepsilon }))^{\frac{q}{2}}\left\Vert
u_{n}\right\Vert ^{q}
\end{equation*
\begin{equation*}
J_{\lambda }(u_{n})\geq \left\Vert u_{n}\right\Vert ^{2}(\frac{N-2}{2N
-\lambda \frac{N-q}{Nq}\Lambda ^{-\frac{q}{2}}V(M)^{1-\frac{q}{N}}(\max
((1+\varepsilon )K_{\circ },A_{\varepsilon }))^{\frac{q}{2}}\left\Vert
u_{n}\right\Vert ^{q-2})
\end{equation*
with $0<\lambda <\frac{\frac{\left( N-2\right) q}{2\left( N-q\right)
\Lambda ^{-\frac{q}{2}}}{V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K_{\circ
},A_{\varepsilon }))^{\frac{q}{2}}\left\Vert u\right\Vert ^{q-2}}$ .

On the other hand, we have
\begin{equation*}
c\geq J_{\lambda }(u_{n})
\end{equation*
\begin{equation*}
\geq \lbrack \frac{N-2}{2N}-\lambda \frac{N-q}{Nq}\Lambda ^{-\frac{q}{2
}V(M)^{1-\frac{q}{N}}(\max ((1+\varepsilon )K_{\circ },A_{\varepsilon }))^
\frac{q}{2}}\left\Vert u_{n}\right\Vert ^{q-2}]\left\Vert u_{n}\right\Vert
^{2}>0
\end{equation*
henc
\begin{equation*}
0\leq \left\Vert u_{n}\right\Vert ^{2}\leq \frac{c}{\frac{N-2}{2N}-\lambda
\frac{N-q}{Nq}\Lambda ^{-\frac{q}{2}}V(M)^{1-\frac{q}{N}}(\max
((1+\varepsilon )K_{\circ },A_{\varepsilon }))^{\frac{q}{2}}\rho ^{q-2}
<+\infty \text{.}
\end{equation*
So $\left( u_{n}\right) _{n}$ is bounded in $H_{2}^{2}(M)$.\ Since \
H_{2}^{2}(M)$ is reflexive and the embedding $H_{2}^{2}(M)\subset \
H_{p}^{k}(M)$ \ ($k\ =0,1$; $p<N$ ) is compact and we have

. $u_{n}\rightarrow u$\ \ weakly in $H_{2}^{2}(M)$.

. $u_{n}\rightarrow u$ strongly in $\ H_{p}^{k}(M)$\ ; $p<N$.

. $u_{n}\rightarrow u$ a.e. in $M$.\newline
The Brezis-Lieb lemma allows us to write
\begin{equation*}
\int_{M}\left\vert \Delta u_{n}\right\vert ^{2}dv_{g}=\int_{M}\left\vert
\Delta u\right\vert ^{2}dv_{g}+\int_{M}\left\vert \Delta
(u_{n}-u)\right\vert ^{2}dvg+o(1)
\end{equation*
and also
\begin{equation*}
\int_{M}f(x)\left\vert u_{n}\right\vert ^{N}dv_{g}=\int_{M}f(x)\left\vert
u\right\vert ^{N}dv_{g}+\int_{M}f(x)\left\vert u_{n}-u\right\vert
^{N}dv_{g}+o(1)\text{.}
\end{equation*
We claim that $u\in M_{\lambda }$, indeed since

$u_{n}\rightarrow u$\ \ weakly in $H_{2}^{2}(M)$, we have for any $\phi \in
H_{2}^{2}(M)$
\begin{equation*}
\int_{M}\left( \Delta u_{n}\Delta \phi -a(x)\left\langle \nabla u_{n},\nabla
\phi \right\rangle +au_{n}\phi \right) \text{ }dv_{g}=
\end{equation*}

\begin{equation*}
\int_{M}\left( \Delta u\Delta \phi -a(x)\left\langle \nabla u,\nabla \phi
\right\rangle +au\phi \right) dv_{g}+o(1)
\end{equation*
and in particular if we let $\phi =u$,
\begin{equation*}
\int_{M}\left( \Delta u_{n}\Delta u-a(x)\left\langle \nabla u_{n},\nabla
u\right\rangle +au_{n}u\right) dv_{g}=\left\Vert u\right\Vert ^{2}+o(1)
\end{equation*
and also if we put $\phi =u_{n}$, we obtai
\begin{equation*}
\int_{M}\left( \Delta u_{n}\Delta u-a(x)\left\langle \nabla u_{n},\nabla
u\right\rangle +au_{n}u\right) dv_{g}=\left\Vert u_{n}\right\Vert ^{2}+o(1
\text{.}
\end{equation*
Now since $(u_{n})_{n}$ belongs to $M_{\lambda }$, we ge
\begin{equation*}
\int_{M}\left( \lambda \left\vert u_{n}\right\vert
^{q-2}u_{n}u+f(x)\left\vert u_{n}\right\vert ^{N-2}u_{n}u\right) dv(g)\
=\left\Vert u\right\Vert ^{2}+o(1)
\end{equation*
and by letting $n\rightarrow +\infty
\begin{equation*}
\int_{M}\left( \lambda \left\vert u_{n}\right\vert
^{q-2}u_{n}u+f(x)\left\vert u_{n}\right\vert ^{N-2}u_{n}u\right)
dv(g)\rightarrow \int_{M}\left( \lambda \left\vert u\right\vert
^{q}+f(x)\left\vert u\right\vert ^{N}\right) dv(g)\text{.}
\end{equation*
Henc
\begin{equation*}
\Phi _{\lambda }(u_{n})=\Phi _{\lambda }(u)=\left\Vert u\right\Vert
^{2}-\lambda \int_{M}\left\vert u\right\vert
^{q}dv(g)-\int_{M}f(x)\left\vert u\right\vert ^{N}dv(g)=0
\end{equation*
and we have
\begin{equation*}
\left\Vert u\right\Vert +o(1)=\left\Vert u_{n}\right\Vert \geq \rho \text{.}
\end{equation*
Consequently $u\in M_{\lambda }$.

Also we claim that $\mu _{n}\rightarrow 0$ as $n$ $\rightarrow +\infty $ in
fact testing with $u_{n}$, we get
\begin{equation*}
\left\langle \nabla J_{\lambda }(u_{n})-\mu _{n}\nabla \Phi _{\lambda
}(u_{n}),u_{n}\right\rangle =o(1)
\end{equation*}

\begin{equation*}
=\underset{=0}{\underbrace{\left\langle \nabla J_{\lambda
}(u_{n}),u_{n}\right\rangle }}-\mu _{n}\left\langle \nabla \Phi _{\lambda
}(u_{n}),u_{n}\right\rangle =o(1)\text{.}
\end{equation*
hence
\begin{equation*}
\mu _{n}\left\langle \nabla \Phi _{\lambda }(u_{n}),u_{n}\right\rangle =o(1)
\end{equation*
and by Lemma \ref{lem3}, we hav
\begin{equation*}
\lim \sup_{n}\left\langle \nabla \Phi _{\lambda }(u_{n}),u_{n}\right\rangle
<0\newline
\end{equation*
so $\mu _{n}\rightarrow 0$ as $n\rightarrow +\infty $.\newline
We are going to show now that $u_{n}\rightarrow u$ converges strongly in
H_{2}^{2}(M)$. First we hav
\begin{equation*}
J_{\lambda }(u_{n})-J_{\lambda }(u)
\end{equation*
\begin{equation}
=\frac{1}{2}\int_{M}\left( \Delta (u_{n}-u)\right) ^{2}dv_{g}-\frac{1}{N
\int_{M}f(x)\left\vert u_{n}-u\right\vert ^{N}dv_{g}+o(1)  \tag{10}
\label{10}
\end{equation
and since $u_{n}-u\rightarrow 0$\ converges weakly in $H_{2}^{2}(M)$, by
testing $\nabla J_{\lambda }(u_{n})-\nabla J_{\lambda }(u)$, we get

\begin{equation*}
\left\langle \nabla J_{\lambda }(u_{n})-\nabla J_{\lambda
}(u),u_{n}-u\right\rangle =o(1)
\end{equation*}

\begin{equation*}
=\int_{M}\left( \Delta (u_{n}-u)\right) ^{2}dv_{g}-\int_{M}f(x)\left\vert
u_{n}-u\right\vert ^{N}dv_{g}=o(1)
\end{equation*
that is to say
\begin{equation}
\int_{M}\left( \Delta _{g}(u_{n}-u)\right) ^{2}dv_{g}=\int_{M}f(x)\left\vert
u_{n}-u\right\vert ^{N}dv_{g}+o(1)  \tag{11}  \label{11}
\end{equation
hence taking account of (\ref{10}), we obtain$\ $

\begin{equation*}
J_{\lambda }(u_{n})-J_{\lambda }(u)=\frac{2}{n}\int_{M}\left( \Delta
(u_{n}-u)\right) ^{2}dv_{g}+o\left( 1\right) .
\end{equation*
The Sobolev inequality allows us to write

\begin{equation*}
\left\Vert u_{n}-u\right\Vert _{N}^{2}\leq (1+\varepsilon )K_{\circ
}\int_{M}\left( \Delta (u_{n}-u)\right) ^{2}dv_{g}+o(1)
\end{equation*
s
\begin{equation}
\int_{M}f(x)\left\vert u_{n}-u\right\vert ^{N}dv_{g}\leq (1+\varepsilon )^
\frac{n}{n-4}}\max_{x\in M}f(x)K_{\circ }^{\frac{n}{n-4}}\left\Vert \Delta
(u_{n}-u)\right\Vert _{2}^{N}+o(1)\text{.}  \tag{12}  \label{12}
\end{equation
Taking account of equality (\ref{11}), one writes

\begin{equation*}
o(1)\geq \left\Vert \Delta (u_{n}-u)\right\Vert _{2}^{2}-(1+\varepsilon )^
\frac{n}{n-4}}\max_{x\in M}f(x)K_{\circ }^{\frac{n}{n-4}}\left\Vert \Delta
(u_{n}-u)\right\Vert _{2}^{N}+o(1)
\end{equation*}

\begin{equation*}
\geq \left\Vert \Delta (u_{n}-u)\right\Vert _{2}^{2}(\left( 1-(1+\varepsilon
)^{\frac{n}{n-4}}\max_{x\in M}f(x)K_{\circ }^{\frac{n}{n-4}}\left\Vert
\Delta (u_{n}-u)\right\Vert _{2}^{N-2}\right) +o(1)\text{.}
\end{equation*
Hence if
\begin{equation*}
\lim \sup_{n}\left\Vert \Delta (u_{n}-u)\right\Vert _{2}^{N-2}<\frac{1}
\left( (1+\varepsilon )K_{o}\right) ^{\frac{n}{n-4}}\max_{x\in M}f(x)}
\end{equation*
we get
\begin{equation*}
\frac{2}{n}\int_{M}\left\vert \Delta _{g}(u_{n}-u)\right\vert ^{2}dv(g)<
\text{.}
\end{equation*
Since
\begin{equation*}
c<\ \frac{2}{n\text{ }K_{\circ }^{\frac{n}{4}}(\max \text{ }_{x\in M}f(x))^
\frac{n-4}{4}}}
\end{equation*
it follows that
\begin{equation*}
\int_{M}\left( \Delta (u_{n}-u)\right) ^{2}dv_{g}<\frac{1}{K_{\circ }^{\frac
n}{4}}(\max_{x\in M}f(x))^{\frac{n-4}{4}}}\text{.}
\end{equation*
Consequently
\begin{equation*}
o(1)\geq \left\Vert \Delta (u_{n}-u)\right\Vert _{2}^{2}\underbrace
(1-(1+\varepsilon )^{\frac{n}{n-4}}\max_{x\in M}f(x)K_{\circ }^{\frac{n}{n-4
}\left\Vert \Delta (u_{n}-u)\right\Vert _{2}^{N-2})}_{>0\newline
}+o(1)
\end{equation*
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ which shows that $\
\begin{equation*}
\left\Vert \Delta (u_{n}-u)\right\Vert _{2}^{2}=o(1)
\end{equation*
and $u_{n}\rightarrow u$ converges strongly in $H_{2}^{2}(M)$.
\end{proof}

\begin{theorem}
\label{th3} Let $\left( M,g\right) $ be an $n$-dimensional compact
Riemannian manifold with $n\geqslant 6$ and $f$ be a smooth positive
function. Assume that the operator $P(u)=\Delta ^{2}u+\bigtriangledown
^{i}\left( a(x)\bigtriangledown _{i}u\right) +b(x)u$ is coercive and
\begin{equation*}
c<\frac{2}{nK_{o}^{\frac{n}{4}}\max_{x\in M}f(x)^{\frac{n}{4}-1}}
\end{equation*}

Then there exists a $\Lambda >0$ such that for any $\lambda \in \left(
0,\Lambda \right) $, equation (\ref{1}) have a non trivial weak solution.
\end{theorem}

\begin{proof}
According to Lemma \ref{lem2}, Lemma \ref{lem3} and Lemma \ref{lem4}, we
infer the existence of $v\in M_{\lambda }$ such that $J_{\lambda
}(v)=\min_{u\in M_{\lambda }}J_{\lambda }$. So there is a real $\mu $ such
that
\begin{equation*}
\bigtriangledown J_{\lambda }(v)=\mu \bigtriangledown Q_{\lambda }(v)
\end{equation*
and multiplying by $v$ and taking account of Lemma\ref{lem3} we obtain that
\mu =0$. Hence $v$ is a non solution of equation (\ref{1}).
\end{proof}

\section{Multiplicity of solutions in case of constant coefficients}

When $P_{g}$ has constant coefficients, we set

\begin{equation*}
J_{\lambda }^{+}(u)=\frac{1}{2}\left( \left\Vert \Delta u\right\Vert
_{2}^{2}-\alpha \int_{M}\left\vert \nabla u\right\vert ^{2}dv_{g}+\beta
\int_{M}u^{2}dv_{g}\right) -\frac{\lambda }{q}\int_{M}u^{+q}dv_{g}\text{\ \
\ }
\end{equation*
\begin{equation*}
-\frac{1}{N}\int_{M}f(x)u^{+N}dv_{g}
\end{equation*
where
\begin{equation*}
u^{+}=\max \left( u,0\right) \text{ }
\end{equation*
Critical points of $J_{\lambda }^{+}$ are solutions t
\begin{equation}
\Delta ^{2}u+\alpha \Delta u+\beta u=\lambda \left( \left( u^{+}\right)
^{q-2}+f\left( u^{+}\right) ^{N-2}\right) u^{+}\text{.}  \tag{13}  \label{13}
\end{equation
Similar arguments as the ones used in the precedent sections give that
J_{\lambda }^{+}$ has a critical point $u$. Standard arguments show that $u$
is of class $C^{4,\theta }$ with $\theta \in \left( 0,1\right) $. If $\alpha
^{2}-4\beta >0$, we let $x_{1}=\frac{\alpha -\sqrt{\alpha ^{2}-4\beta }}{2}$
and $x_{2}=\frac{\alpha +\sqrt{\alpha ^{2}-4\beta }}{2}$ \ and moreover if
\alpha >0$, then $x_{1}$, $x_{2}>0$ and

\begin{equation*}
\left( \Delta +x_{1}\right) \left( \Delta +x_{2}\right) u=\Delta
^{2}u+\alpha \Delta u+\beta u\geq 0\text{.}
\end{equation*
Applying the maximum principle twice, we obtain that $u$ is a positive
solution of class $C^{4,\theta }$, where $\theta \in (0,1)$ of the equatio
\begin{equation*}
\Delta ^{2}u+\alpha \Delta u+\beta u=\lambda \left( u^{q-1}+fu^{N-1}\right)
\text{.}
\end{equation*
and standard regularity results give that $u$ is smooth.

In the same manner if we set

\begin{equation*}
J_{\lambda }^{-}(u)=\frac{1}{2}\left( \left\Vert \Delta u\right\Vert
_{2}^{2}-\alpha \int_{M}\left\vert \nabla u\right\vert ^{2}dv_{g}+\beta
\int_{M}u^{2}dv_{g}\right) -\frac{\lambda }{q}\int_{M}\left\vert
u^{-q}\right\vert dv_{g}\text{\ \ \ }
\end{equation*
\begin{equation*}
-\frac{1}{N}\int_{M}f(x)\left\vert u^{-}\right\vert ^{N}dv_{g}
\end{equation*
where
\begin{equation*}
u^{-}=\min \left( u,0\right) \text{ }
\end{equation*
then the critical points of $J_{\lambda }^{-}$ are solutions t
\begin{equation*}
\Delta ^{2}u+\alpha \Delta u+\beta u=\lambda \left( \left\vert
u^{-}\right\vert ^{q-2}+f\left\vert u^{-}\right\vert ^{N-2}\right) u^{-
\text{.}
\end{equation*
By the \ same argument as above we get that $u^{-}$ is a negative smooth
solution. Similar arguments as the ones we used for $J_{\lambda }$ give that
$J_{\lambda }^{+}$ and $J_{\lambda }^{-}$ have critical points $M_{\lambda
}^{+}$ and $M_{\lambda }^{-}$ respectively where
\begin{equation*}
M_{\lambda }^{\pm }=\left\{ u\in H_{2}:Q_{\lambda }^{\pm }\left( u\right) =
\text{ and }\left\Vert u\right\Vert \geq \rho >0\right\}
\end{equation*
an
\begin{equation*}
Q_{\lambda }^{\pm }\left( u\right) =\left\langle \bigtriangledown J_{\lambda
}^{\pm }(u),u\right\rangle \text{.}
\end{equation*}

Summarizing, we get

\begin{theorem}
\label{th4} Let $\left( M,g\right) $ be an $n$-dimensional compact
Riemannian manifold with $n\geqslant 6$. Assume that the operator
P(u)=\Delta ^{2}u+\bigtriangledown ^{i}\left( a(x)\bigtriangledown
_{i}u\right) +b(x)u$ is coercive and
\begin{equation*}
c<\frac{2}{nK_{o}^{\frac{n}{4}}\max_{x\in M}f(x)^{\frac{n}{4}-1}}\text{.}
\end{equation*}

If moreover $\frac{\alpha ^{2}}{4}>\beta $ with $\alpha >0$. Then equation
\ref{5}) has two distinct smooth solutions; one positive and the other
negative.
\end{theorem}

\begin{lemma}
\label{lem6} For any $\lambda >0$, sufficiently small, $J_{\lambda }$ has
two local minima.
\end{lemma}

\begin{proof}
We follow closely the proof of Lemma 8 in \cite{2}. As a consequence of
Lemma \ref{lem2}, Lemma \ref{lem3} and Lemma \ref{lem5}, we infer the
existence of $v_{1}\in M_{\lambda }^{+}$ and a $v_{2}\in M_{\lambda }^{-}$
such that
\begin{equation*}
J_{\lambda }^{+}(v_{1})=\min_{u\in M_{\lambda }^{+}}J_{\lambda }^{+}(u)
\end{equation*
an
\begin{equation*}
J_{\lambda }^{-}(v_{2})=\min_{u\in M_{\lambda }^{-}}J_{\lambda }^{-}(u)\text
.}
\end{equation*
Note that $v_{1}$ and $v_{2}$ are respectively smooth positive and negative
solutions of equation (\ref{5}). Indeed by Lagrange mutiplicators theorem we
get that
\begin{equation*}
\bigtriangledown J_{\lambda }^{+}\left( v_{1}\right) =\mu \bigtriangledown
Q_{\lambda }^{+}(v_{1})
\end{equation*
and multiplying by $v_{1}$ we deduce tha
\begin{equation*}
\mu =0
\end{equation*
Hence $v_{1}$ is a solution of (\ref{13}) and as in section 3 we get that
v_{1}$ positive, hence a positive solution of equation ( \ref{5}). $v_{2}$
is actually a negative solution of (\ref{5}). We claim that $v_{1}$ and
v_{2}$ are local minima of $J_{\lambda }$ if it is not the case let
w_{n}\in M_{\lambda }$ such that $w_{n}\rightarrow v_{1}$ in $H_{2}^{2}$ as
n\rightarrow +\infty $ and
\begin{equation}
J_{\lambda }(w_{n})<J_{\lambda }^{+}\left( v_{1}\right)  \tag{14}  \label{14}
\end{equation
We can choose $w_{n}$ as
\begin{equation}
J_{\lambda }(w_{n})=\inf_{u\in B_{n}\cap M_{\lambda }}J_{\lambda }(u)
\tag{15}  \label{15}
\end{equation
where $B_{n}=\left\{ u\in \left\Vert u-v_{1}\right\Vert _{H_{1}^{2}}\leq
\frac{1}{n}\right\} $. There exist parameters $\lambda _{n}$ and $\mu _{n}$
such tha
\begin{equation}
\bigtriangledown J_{\lambda }(w_{n})=\lambda _{n}\bigtriangledown Q_{\lambda
}(w_{n})+\mu _{n}\left( \Delta ^{2}w_{n}+\alpha \Delta w_{n}+\beta
w_{n}\right)  \tag{16}  \label{16}
\end{equation
with $\mu _{n}\leq 0$. Taking the inner product of the latter equality with
w_{n}$, we ge
\begin{equation*}
\lambda _{n}\left\langle \bigtriangledown Q_{\lambda
}(w_{n}),w_{n}\right\rangle +\mu _{n}\left\Vert w_{n}\right\Vert
_{H_{1}^{2}}^{2}=0
\end{equation*
and we infer that $\lambda _{n}\leq 0$.

Equation reads as
\begin{equation*}
\Delta ^{2}w_{n}+\alpha \Delta w_{n}+\beta w_{n}=\frac{\left( -\lambda
_{n}-\mu _{n}\right) }{\left( 1-\lambda _{n}-\mu _{n}\right) }f\left\vert
w_{n}\right\vert ^{N-2}w_{n}\text{.}
\end{equation*
By standard methods, $w_{n}$ is of class $C^{4,\theta }$ , $0<\theta <1$.
Hence $w_{n}$ goes to $v_{1}$ in the $C^{2}$ topology, then $w_{n}>0$. So
\ref{15}) is a contradiction with (\ref{14}). Hence $v_{1}$ and $v_{2}$ are
respectively positive and negative solution of equation (\ref{5}) of minimal
positive energy.
\end{proof}

Next we prove

\begin{theorem}
\label{th5} Let $\left( M,g\right) $ be an $n$-dimensional compact
Riemannian manifold with $n\geqslant 6$. Assume that the operator
P(u)=\Delta ^{2}u+\bigtriangledown ^{i}\left( a(x)\bigtriangledown
_{i}u\right) +b(x)u$ is coercive and
\begin{equation*}
c<\frac{2}{nK_{o}^{\frac{n}{4}}\max_{x\in M}f(x)^{\frac{n}{4}-1}}\text{.}
\end{equation*}

If moreover $\frac{\alpha ^{2}}{4}>\beta $ with $\alpha >0$. Then equation
\ref{5}) has third solution $w$ distinct of $u^{+}$ and $u^{-}$.
\end{theorem}

\begin{proof}
We can suppose that the minima of $J_{\lambda }$ are realized by $u^{+}$ and
$u^{-}$ . The geometric conditions of the Mountain pass theorem are
satisfied. If $\Gamma $ denotes the set of paths $\gamma :\left[ 0,1\right]
\rightarrow M_{\lambda }$ such that $\gamma (0)=u^{-}$ and $\gamma (1)=u^{+}
. Let $c_{\lambda }=\inf_{\gamma \in \Gamma }\max_{t\in \left[ 0,1\right]
}\left( J_{\lambda }\left( \gamma )\right) \right) $. By Lemma \pageref{4},
we infer that $c_{\lambda }$ is a critical level of the function $J_{\lambda
}$ with critical value $w$ and by Lemma \ref{3} $w\in M_{\lambda }$. Hence
w $ is solution of equation (\ref{5}) different from $u^{+}$ and $u^{-}$.
\end{proof}

\section{Test functions}

In this section we give the proof of Theorem \ref{th1} and \ref{th2}.
\newline
$Let $(y^{1},...,y^{n})$ be normal coordinates centred at the point
x_{\circ }$ where the function attains its maximum and $S(r)$ be the
geodesic sphere centred at $x_{o}$ and of radius $r$ ( $r<d$ the injectivity
radius ). Denote by $d\sigma $ the volume element of the $n-1$-dimensional
unit $S^{n-1}$.

Put
\begin{equation*}
G(r)=\frac{1}{w_{n-1}}\dint\limits_{S(r)}\sqrt{\left\vert g(x)\right\vert
d\sigma
\end{equation*
where $w_{n-1}$ denotes the area of $S^{n-1}$ and $\left\vert
g(x)\right\vert $ the determinant of the metric $g$. An expansion of $G(r)$
in a neighborhood of $r=0$ writes as
\begin{equation*}
G(r)=1-\frac{S_{g}(x_{\circ })}{6n}r^{2}+o(r^{2})
\end{equation*
where $S_{g}(x_{\circ })$ denotes the scalar curvature of $M$ at the point
x_{\circ }$.$\newline
$Let $B(x_{\circ },\delta )$ be the ball centred at $x_{\circ }$ and of
radius $\delta $ with $0<2\delta <d$ and let $\eta $ be a smooth function
equals to $1$ on $B\left( x_{o},\delta \right) $ and equals to $0$ on
M-B\left( x_{o},2\delta \right) $.$\newline
$ Put

\begin{equation*}
u_{\epsilon }(x)=(\frac{(n-4)n(n^{2}-4)\epsilon ^{4}}{f(x_{\circ })})^{\frac
n-4}{8}}\frac{\eta (r)}{(r^{2}+\epsilon ^{2})^{\frac{n-4}{2}}}
\end{equation*
where \
\begin{equation*}
f(x_{\circ })=\max_{x\in M}f(x)
\end{equation*
and $r=d(x_{\circ },.)$ is geodesic distance to the point $x_{\circ }$
\newline
We let, for $p-q>1$,

\begin{equation*}
I_{p}^{q}=\int_{0}^{+\infty }\frac{t^{q}}{(1+t)^{p}}dt
\end{equation*
which fulfills
\begin{equation*}
I_{p+1}^{q}=\frac{p-q-1}{p}I_{p}^{q}\text{ \ \ \ and \ \ \ \ }I_{p+1}^{q+1}
\frac{q+1}{p-q-1}I_{p+1}^{q}\text{.}
\end{equation*
In the case where the dimension of the manifold $n>6$, we have

\begin{theorem}
Let $\left( M,g\right) $ be an $n$-dimensional compact Riemannian manifold
with $n>6$. If at the point $x_{o}$ where the function $f$ achieves its
maximum
\begin{equation*}
\frac{n\left( n^{2}+4n-20\right) }{2\left( n-2\right) \left( n^{2}-4\right)
S_{g}\left( x_{o}\right) +\frac{n\left( n-6\right) }{\left( n-2\right)
\left( n^{2}-4\right) }a\left( x_{o}\right) -\frac{n}{8\left( n-2\right)
\frac{\Delta f(x_{o})}{f\left( x_{o}\right) }>0
\end{equation*}
equation (\ref{1}) have a non trivial solution of class $C^{4,\alpha }(M)$,
\alpha \in (0,1)$.
\end{theorem}

\begin{proof}
As in (\cite{7}), we get
\begin{equation*}
\int_{M}f(x)\left\vert u_{\epsilon }(x)\right\vert ^{N}dv(g)=\frac{1}
K_{\circ }^{\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4}}}\left( 1-(\frac
\Delta f(x_{\circ })}{2(n-2)f(x_{\circ })}+\frac{S_{g}(x_{\circ })}{6(n-2)
)\epsilon ^{2}+o(\epsilon ^{2})\right)
\end{equation*
and als
\begin{equation*}
\int_{M}a(x)\left\vert \nabla u_{\epsilon }\right\vert ^{2}dv(g)=\frac{1}
K_{\circ }^{\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4}}}\left( \frac
4(n-1)a(x_{\circ })}{(n^{2}-4)(n-6)}\epsilon ^{2}+o(\epsilon ^{2})\right)
\text{.}
\end{equation*
The computations giv
\begin{equation*}
\int_{M}b(x)u_{\epsilon }^{2}dv(g)=o(\epsilon ^{2})
\end{equation*
an
\begin{equation*}
\int_{M}\left\vert \Delta u_{\epsilon }\right\vert ^{2}dv(g)=\frac{1}
K_{\circ }^{\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4}}}\left( 1-\frac
n^{2}+4n-20}{6(n^{2}-4)(n-6)}S_{g}(x_{\circ })\epsilon ^{2}+o(\epsilon
^{2})\right) \text{.}
\end{equation*
Summarizing we obtai
\begin{equation*}
\int_{M}\left\vert \Delta u_{\epsilon }\right\vert ^{2}-a(x)\left\vert
\nabla u_{\epsilon }\right\vert ^{2}+b(x)u_{\epsilon }^{2}dv(g)=\frac{1}
K_{\circ }^{\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4}}}
\end{equation*
\begin{equation*}
\left( 1-(\frac{n^{2}+4n-20}{6(n^{2}-4)(n-6)}S_{g}(x_{\circ })+\frac{4(n-1)}
(n^{2}-4)(n-6)}a(x_{\circ }))\epsilon ^{2}+o(\epsilon ^{2})\right) \text{.}
\end{equation*
Taking in mind that
\begin{equation*}
J_{\lambda }\left( u_{\epsilon }\right) =\frac{1}{2}\left\Vert u_{\epsilon
}\right\Vert ^{2}-\frac{\lambda }{q}\left\Vert u_{\epsilon }\right\Vert
_{q}^{q}-\frac{1}{N}\int_{M}f(x)\left\vert u_{\epsilon }(x)\right\vert
^{N}dv(g)
\end{equation*
where
\begin{equation*}
\left\Vert u_{\epsilon }\right\Vert ^{2}=\int_{M}\left\vert \Delta
u_{\epsilon }\right\vert ^{2}-a(x)\left\vert \nabla u_{\epsilon }\right\vert
^{2}+b(x)u_{\epsilon }^{2}dv(g)
\end{equation*
and since $\lambda >0$, we get

\begin{equation*}
J_{\lambda }\left( u_{\epsilon }\right) \leq \frac{1}{2}\left\Vert
u_{\epsilon }\right\Vert ^{2}-\frac{1}{N}\int_{M}f(x)\left\vert u_{\epsilon
}(x)\right\vert ^{N}dv(g)
\end{equation*
\begin{equation*}
\leq \frac{1}{K_{\circ }^{\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4}}}\left[
\frac{2}{n}-\left( \frac{n^{2}+4n-20}{(n^{2}-4)(n-6)}S_{g}(x_{\circ })+\frac
2(n-1)}{(n^{2}-4)(n-6)}a(x_{\circ })-\frac{1}{4(n-2)}\frac{\Delta f(x_{\circ
})}{f(x_{\circ })}\right) \epsilon ^{2}+o(\epsilon ^{2})\right]
\end{equation*
\begin{equation*}
\leq \frac{2}{n\text{ }K_{\circ }^{\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4
}}\left[ 1-\left( \frac{\left( n^{2}+4n-20\right) n}{2(n^{2}-4)(n-6)
S_{g}(x_{\circ })+\frac{(n-1)\text{ }n}{(n^{2}-4)(n-6)}a(x_{\circ })-\frac{
}{8(n-2)}\frac{\Delta f(x_{\circ })}{f(x_{\circ })}\right) \epsilon
^{2}+o(\epsilon ^{2})\right] \text{.}
\end{equation*
So the condition
\begin{equation*}
J_{\lambda }\left( u_{\epsilon }\right) <\ \frac{2}{n\text{ }K_{\circ }^
\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4}}}
\end{equation*
is fulfilled if
\begin{equation*}
\left( \frac{\left( n^{2}+4n-20\right) n}{2(n^{2}-4)(n-6)}S_{g}(x_{\circ })
\frac{(n-1)\text{ }n}{(n^{2}-4)(n-6)}a(x_{\circ })-\frac{n}{8(n-2)}\frac
\Delta f(x_{\circ })}{f(x_{\circ })}\right) >0\text{.}
\end{equation*}
\end{proof}

In the case $n=6$, we have

\begin{proof}
The same calculations as in case $n>6$ lead to \
\begin{equation}
\int_{M}f(x)\left\vert u_{\epsilon }(x)\right\vert ^{N}dv(g)=\frac{1}
K_{\circ }^{\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4}}}\left( 1-(\frac
\Delta f(x_{\circ })}{2(n-2)f(x_{\circ })}+\frac{S_{g}(x_{\circ })}{6(n-2)
)\epsilon ^{2}+o(\epsilon ^{2})\right) \text{.}  \tag{17}  \label{17}
\end{equation
Also the same computations as in \cite{7} with minor modifications allow us
to writ
\begin{equation*}
\int_{M}a(x)\left\vert \nabla u_{\epsilon }\right\vert ^{2}dv(g)=(n-4)^{2}
\frac{(n-4)n(n^{2}-4)}{f(x_{\circ })})^{\frac{n-4}{4}}\frac{w_{n-1}}{2
\left( a(x_{\circ })\epsilon ^{2}\log (\frac{1}{\epsilon ^{2}})\text{
+O(\epsilon ^{2})\right)
\end{equation*
and
\begin{equation*}
\int_{M}\left\vert \Delta u_{\epsilon }\right\vert ^{2}dv(g)=(n-4)^{2}(\frac
(n-4)n(n^{2}-4)}{f(x_{\circ })})^{\frac{n-4}{4}}\frac{w_{n-1}}{2}
\end{equation*
\begin{equation*}
\left( \frac{n(n+2)(n-2)}{(n-4)}I_{n}^{\frac{n}{2}-1}-\frac{2}{n
S_{g}(x_{\circ })\epsilon ^{2}\log (\frac{1}{\epsilon ^{2}})+O(\epsilon
^{2})\right) \text{.}
\end{equation*
Consequentl
\begin{equation*}
\int_{M}\left( \Delta u_{\epsilon }\right) ^{2}-a(x)\left\vert \nabla
u_{\epsilon }\right\vert ^{2}+b(x)u_{\epsilon }^{2}dv(g)=(n-4)^{2}(\frac
(n-4)n(n^{2}-4)}{f(x_{\circ })})^{\frac{n-4}{4}}\frac{w_{n-1}}{2}
\end{equation*
\begin{equation*}
\left[ \frac{n(n+2)(n-2)}{(n-4)}I_{n}^{\frac{n}{2}-1}-\left( \frac{2}{n
S_{g}(x_{\circ })+a(x_{\circ })\right) \epsilon ^{2}\log (\frac{1}{\epsilon
^{2}})+O(\epsilon ^{2})\right]
\end{equation*
\begin{equation*}
=\frac{1}{K_{\circ }^{\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4}}}\left( 1
\frac{\left( n-4\right) }{n\left( n^{2}-4\right) I_{n}^{\frac{n}{2}-1}
\left( \frac{2}{n}S_{g}(x_{\circ })+a(x_{\circ })\right) \epsilon ^{2}\log
\frac{1}{\epsilon ^{2}})+O(\epsilon ^{2})\text{ \ }\right)
\end{equation*
and taking account of (\ref{17}), we obtai
\begin{equation*}
J_{\lambda }\left( u_{\epsilon }\right) \leq \frac{1}{2}\left\Vert
u_{\epsilon }\right\Vert ^{2}-\frac{1}{N}\int_{M}f(x)\left\vert u_{\epsilon
}(x)\right\vert ^{N}dv(g)
\end{equation*}

\begin{equation*}
\leq \frac{1}{K_{\circ }^{\frac{n}{4}}(f(x_{\circ }))^{\frac{n-4}{4}}}\left(
\frac{1}{2}-\frac{1}{N}-\frac{\left( n-4\right) t_{o}^{2}}{2n\left(
n^{2}-4\right) I_{n}^{\frac{n}{2}-1}}\left( \frac{2}{n}S_{g}(x_{\circ
})+a(x_{\circ })\right) \epsilon ^{2}\log (\frac{1}{\epsilon ^{2}
)+O(\epsilon ^{2})\right) \text{.}
\end{equation*
So if in the point $x_{o}$ where the maximum of the function $f$ is
achieved, the condition $\frac{2}{n}S_{g}(x_{\circ })+a(x_{\circ })>0$ i.e.
since $n=6$, $S_{g}(x_{\circ })>-3a(x_{\circ })$ is fulfilled, we get for
\epsilon $ sufficiently smal
\begin{equation*}
J_{\lambda }\left( u_{\epsilon }\right) <\frac{2}{nK_{\circ }^{\frac{n}{4
}(f(x_{\circ }))^{\frac{n-4}{4}}}\text{.}
\end{equation*}
\end{proof}

redpajama_set_name: RedPajamaArXiv


Outside of the return of Caris LeVert it was a night to forget for the Brooklyn Nets. They went down heavily to the Chicago Bulls in one of their worst performances of the season. Here's how the loss unfolded.
D'Angelo Russell with the and one!


When I was growing up, when you had to enter a code on the phone, it was always followed by the Pound Sign. So, now when I see women posting things or news about #metoo, my mind immediately goes to "Why would she want someone to pound her? Isn't that what they are protesting?"


Soapy clothes - socks, shirts, bras, and underwear – are pressed up against the glass before they slide deeper into the wash drum.
What did you say before you took your clothes and books out of our apartment?
The soap scrubs it clean – taking a little of your essence from me - even as it bleeds onto the rest of my clothes.
I feel a little melancholy doing laundry alone.


We warmly welcome you to our easter-special in our online shop.
In our assortment you will find a fine selection of wireless devices, which are characterized by excellent coverage, high data rates and a good operability.
We specialize in solutions of general problems relating to Wi-Fi, such as disconnections at large distances or a weak wireless signal in buildings.
In addition, we also offer tailor-made systems for more demanding applications such as hotel-hotspot-systems, Wi-Fi while traveling, and the wireless networking between buildings and company premises.
Besides a good selection of high-performance devices with an excellent price-performance ratio, we offer a qualified and comprehensive advisory service for private and commercial wireless applications.
As a specialist trading company with in house planning- and projecting-office, we use the products we sell in our daily work. Due to our long experience we know exactly what is important.
Do you have any questions or suggestions about our products or services?


text:
NeverAlone.Love
CHOPRA FOUNDATION INITIATIVE
Never Alone Initiative – A Worldwide Alliance
Never Alone is forging an alliance of people and organizations dedicated to incubating a new vision for mental and emotional health and the resources available to people seeking support.
Creating a Worldwide Conversation to Reframe Mental Health
The Never Alone Alliance represents a collaboration of businesses, policy makers, mental health professionals, wellness initiatives, scientists, schools and more. Our goal – build mental health awareness, advance scientific research, work with policy-makers and create a global technology platform to democratize access to resources.
Deepak Chopra, MD, FACP
The Chopra Foundation Founder, Chairman of the Board and Director
Deepak Chopra MD, FACP, founder of The Chopra Foundation and Chopra Global, is a world-renowned pioneer in integrative medicine and personal transformation. He is a Clinical Professor of Family Medicine and Public Health at the University of California, San Diego. His 90th book and national bestseller, Metahuman: Unleashing Your Infinite Potential (Harmony Books), unlocks the secrets to moving beyond our present limitations to access a field of infinite possibilities. TIME magazine has described him as "one of the top 100 heroes and icons of the century."
Gabriella Wright
Co-Founder, Never Alone Initiative Actress, Humanitarian and Activist
Gabriella Wright is a professional actress, model, activist, and motivational speaker with a long-standing humanitarian engagement. A co-founder of the Never Alone Initiative, she is the innovator of the Mental Hygiene Toolkit, a collection of mind cleansing and self-awareness tools to help individuals nurture their bodies, minds, and spirits, and develop inner guidance to consciously choose the best path for their lives. Wright has lent her voice to UN Women and the UN Trust Fund to End Violence against Women. Among many other societal contributions, she is a Global Goodwill Ambassador to Skypower and the Honorary President of a grassroots charity in Myanmar.
Poonacha Machaiah
Co-founder, Never Alone Initiative Founder, Warrior Monk
Poonacha Machaiah is a global thought leader with unique abilities to combine the wellbeing and technology domains. Among a new breed of social entrepreneurs, he applies his experience from 25 years as a business executive in Fortune 100 companies and entrepreneurial initiatives to address societal and wellbeing challenges. This led him to co-found the Never Alone initiative, as well as launch the Warrior Monk brand targeted at creating a positive societal shift through the compassionate transformation of humankind. He formerly held senior management positions at Nortel, Iridium, Motorola and Sasken.
Claudio Gutierrez
Founder and CEO of CG Creative Studios and the Marketing/Communications Alliance Partner of the Never Alone Initiative
Claudio Gutierrez has been breaking barriers in audience engagement, event communications, brand strategy and multimedia design for the past two decades. He began his career in branding, design and advertising before immersing himself in video/film production and corporate events, and then ultimately, delivering brand strategy and experiences to companies like Hewlett Packard Enterprise, Microsoft, Amazon, Sony, Cisco, Hewlett-Packard, Hubspot, Micro Focus, Rubrik and Hedera Hashgraph. He is a co-founder and trustee of the Inspire Dialogue Foundation, based in Cambridge, UK., which hosted the Dalai Lama.
Naveen Jain
Founder & CEO, Viome
Francois Ferre
Life Sciences Entrepreneur, Investor, and Board member. Genomics and Diagnostics. AltheaDX, Alma Life Sciences
Paul J. Mills
Professor of Family Medicine and Public Health, University of California, San Diego

redpajama_set_name: RedPajamaCommonCrawl


Crossword Puzzle Helper - Software Informer. A neat and useful tool designed to help you when playing crossword puzzle games.
Crossword Puzzle Helper was developed to find words with a certain length when you do not know all the letters. Type in the search word using a wild card (? or /) for each unknown letter and press the "Start" button. All words meeting the search criteria will be displayed in the table. To search for another word, simply enter a new search word and click on the "Start" button. The table will be cleared and the new results will be displayed.


text:
Posted on November 28, 2021 by Steven Hayward in Academic left, Identity Politics
Elizabeth Warren, Call Your Office
Cast your mind back to 2014 for a moment, when the St. Louis Post-Dispatch newspaper canceled George Will's column because of this passage:
Colleges and universities are being educated by Washington and are finding the experience excruciating. They are learning that when they say campus victimizations are ubiquitous ("micro-aggressions," often not discernible to the untutored eye, are everywhere), and that when they make victimhood a coveted status that confers privileges, victims proliferate.
Of course you're not allowed to say that, because as we all know only "white privilege" exists, oppressing everything and everybody.
And yet we have still another fresh case of someone claiming membership in an oppressed class to get ahead, this time in Canada:
She's Sitting Bulls-t.
A Canadian medical researcher who rose to become the nation's top voice on indigenous health has been ousted from her government job and her university professorship — after suspicious colleagues investigated her increasingly fanciful claims of Native American heritage and learned she was a fraud.
Carrie Bourassa, a public health expert who served as scientific director of the Canadian Institutes of Health Research's Institute of Indigenous Peoples' Health, was suspended on Nov. 1, five days after the state-owned Canadian Broadcasting Corporation published a lengthy expose on her background.
Far from being a member of the Métis nation, as she had long claimed, a laborious trace of Bourassa's family tree revealed that her supposedly indigenous ancestors were in fact immigrant farmers who hailed from Russia, Poland, and Czechoslovakia.
She was just following Elizabeth Warren's warpath:
Leftists: Fakers and frauds all the way down.


A leading North Wales and Shropshire law firm has gained accreditation in the Law Society's Wills and Inheritance Quality Scheme (WIQS).
Q: My first wife and I left everything to each other in our Wills, but after she died I made a new Will leaving everything to our three sons. Now I am planning to marry again. My new partner also has grown-up children who are the sole beneficiaries of her Will. We plan to live in my house after we are married and rent her house out, so if anything happens to one of us the other can revert to living how we did before. Do we need to update our Wills to ensure that our respective children will remain sole beneficiaries of our respective Wills as they are now?
Q: My business partner and I have had five very successful years and we could not have got where we are without the hard work of a dedicated and loyal team of staff. Now we are looking for a cost-effective way of rewarding our employees. In particular we are considering an Enterprise Management Incentives scheme. Are these popular among SMEs?
Q: I have outgrown my leased business premises and want to move. There is still almost two years left to run on my lease, but I have a friend who is keen to take over the premises and the lease. If I assign the lease to him will it exempt me from future liability for the premises?
Q: My mother made an enquiry about stair lifts that were advertised in a magazine. She does not need one but thought she would enquire for the future. A salesman later turned up on her doorstep, talked his way into the house to check the suitability of her staircase for installing a lift, and the next thing was she had signed an order form.
Q: I am in my late 50's and have discovered that my husband has been cheating on me for years. The relationship is over, but I am terrified to divorce him in case I can't manage financially. I gave up my own career to bring up the children and look after my husband's needs and now I would not have the confidence to work. In fact I have no confidence to do anything anymore and I am on medication for stress. What can I do?
Solicitors from a leading North Wales law firm will be on hand to answer questions about Wills and Powers of Attorney at a meeting being held in Wrexham next week for older carers.
Offices to Let in The Old Bank, Berwyn Street, Llangollen LL20 8ND - a highly prominent Grade II Listed building in the heart of Llangollen close to all amenities.


Q. Why should I prepare for entrance?
Q. How is entrance different from boards exams?
Ans. Entrance exam has never before seen question. Most of the topics are unrelated to school curriculum. Pattern of exam changes every year.
Q. When to start preparation?
Ans. The best time to prepare is now!! Register for free e-learning here.


The 12-county Piedmont Triad Region of North Carolina is anchored by the cities of Greensboro, Winston-Salem and High Point in the northwest part of the state. The area is known for its business development as well as a growing arts community. Travelers can find a number of steak and seafood restaurants in the area with varied offerings.
Ryan's Restaurant (ryansrestaurant.com) opened its doors in 1977 and is located just minutes from Wake Forest University in Winston-Salem. The fine-dining restaurant offers a fireplace in winter and a view of oak trees and a rolling stream from its large windows year round. The 16-ounce prime New York strip steak tops the beef menu and the live Maine lobster is a house specialty. There is a full bar and extensive wine list.
Longhorn Steakhouse (longhornsteakhouse.com) offers steaks and seafood entrees but specializes in sandwiches and salads with meat and fish as well. House signature dishes include the grilled trout topped with shrimp and lobster and the 20-ounce porterhouse. The family friendly steakhouse offers a kids menu and is located in Greensboro.
Two Japanese restaurants in Winston-Salem cook hibachi-style, where the food is grilled in front of guests. At Kanpai Japanese Steak & Seafood House (kanpaiws.com), all the dinner entrees are served with a flaming shrimp appetizer, salad, vegetables and fried rice. Steak and seafood combination plates include shrimp and ribeye steak, fillet mignon and scallops and fillet and lobster.
At Arigato Japanese Steakhouse (arigatos.net), guests can order steak, seafood or combinations served with broth, salad, shrimp appetizers, vegetables and fried rice. Specialtiy combinations include lobster and fillet mignon or shrimp and ribeye.
Both restaurants have full bars.
pasta, the Cajun shrimp and crab cakes. On the beef menu guests find a ribeye, New York strip and filet mignon. The restaurants has a full bar and is in High Point.
Oso Tavern (osorestaurantgroup.com) offers lunch and dinner in High Point in a friendly atmosphere. The brick-and-wood interior restaurant has a bar that offers cocktails, an extensive wine list and numerous beers on tap. Little plate choices include Thai shrimp, Cajun-fried oysters and beef brushchetta. Entrees vary from standard grilled steaks to offerings such as a grilled, chopped steak with marinara sauce or beef tips and mushrooms, seared sea bass and salmon with capers.
Teubner, Nellene. "Steak & Seafood Restaurants in the Piedmont Triad Region, North Carolina." Travel Tips - USA Today, https://traveltips.usatoday.com/steak-seafood-restaurants-piedmont-triad-region-north-carolina-63587.html. Accessed 21 April 2019.


text:
Q: Modify arguments for parent theme's `wp_register_style` via child theme Working on a child theme for site with a good deal of separate stylesheets being registered/enqueued from the theme and plugins. I'd like to change some attributes of the <link> tags in the final output, while avoiding conflicts after a parent theme update by the site owner.
The goal is to not re-register/re-enqueue the same .css files already set by the parent theme and/or plugins, while effectively passing a new set of arguments to the wp_register_style function.
For example, currently this is what the parent theme & plugins are doing:
// parent theme or plugin's php
wp_register_style( 'a_stylesheet', $uri . 'css/a-stylesheet.css', array(), $version, 'screen' );

// html output
<link rel='stylesheet' id='a_stylesheet-css'  href='http://domain.com/wp-content/themes/theme/css/a-stylesheet.css' type='text/css' media='screen' />

The goal is for the child theme to in effect change the wp_register_style call, like so:
// child theme's php
some_function_to_reregister( 'already_registered_file', $inherit, $inherit, $inherit, 'handheld' )
// if the above doesn't exist, do nothing; else, re-register with the new argument(s).

// html output
<link rel='stylesheet' id='a_stylesheet-css'  href='http://domain.com/wp-content/themes/theme/css/a-stylesheet.css' type='text/css' media='handheld' />

This changes the <link media> attribute to handheld instead of screen, but preserves everything else being done by the parent theme/plugin, and skips registering the file if it hasn't been registered yet elsewhere (i.e. after an update).
Seems like this would be a much more efficient approach than de-registering/de-enqueuing css from the parent theme and plugins only to copy/paste it back in the child theme and have unexpected wreckage in future updates.

A: I don't think there's a built-in safe way to do this, but you could directly modify the $wp_styles global after things are registered but before they're output, or at least use it to fetch the parameters a style was originally registered with.
global $wp_styles;
if( isset( $wp_styles->registered['a_stylesheet'] ) ){
    $wp_styles->registered['a_stylesheet']->args = 'handheld';
}

Not ideal, but it may work for you.

redpajama_set_name: RedPajamaStackExchange


text:
Technology|In Fight Over Piracy Bills, New Economy Rises Against Old
In Fight Over Piracy Bills, New Economy Rises Against Old
By JONATHAN WEISMAN JAN. 18, 2012
The Web sites on Wednesday of, clockwise from top left, Google, Mozilla, Wired and Wikipedia.
WASHINGTON — When the powerful world of old media mobilized to win passage of an online antipiracy bill, it marshaled the reliable giants of K Street — the United States Chamber of Commerce, the Recording Industry Association of America and, of course, the motion picture lobby, with its new chairman, former Senator Christopher J. Dodd, the Connecticut Democrat and an insider's insider.
Yet on Wednesday this formidable old guard was forced to make way for the new as Web powerhouses backed by Internet activists rallied opposition to the legislation through Internet blackouts and cascading criticism, sending an unmistakable message to lawmakers grappling with new media issues: Don't mess with the Internet.
As a result, the legislative battle over two once-obscure bills to combat the piracy of American movies, music, books and writing on the World Wide Web may prove to be a turning point for the way business is done in Washington. It represented a moment when the new economy rose up against the old.
"I think it is an important moment in the Capitol," said Representative Zoe Lofgren, Democrat of California and an important opponent of the legislation. "Too often, legislation is about competing business interests. This is way beyond that. This is individual citizens rising up."
It appeared by Wednesday evening that Congress would follow Bank of America, Netflix and Verizon as the latest institution to change course in the face of a netizen revolt.
Legislation that just weeks ago had overwhelming bipartisan support and had provoked little scrutiny generated a grass-roots coalition on the left and the right. Wikipedia made its English-language content unavailable, replaced with a warning: "Right now, the U.S. Congress is considering legislation that could fatally damage the free and open Internet." Visitors to Reddit found the site offline in protest. Google's home page was scarred by a black swatch that covered the search engine's label.
Phone calls and e-mail messages poured in to Congressional offices against the Stop Online Piracy Act in the House and the Protect I.P. Act in the Senate. One by one, prominent backers of the bills dropped off.
Senator Marco Rubio co-sponsored the Protect I.P. Act, but he renounced it on Wednesday. Credit Philip Scott Andrews/The New York Times
First, Senator Marco Rubio of Florida, a rising Republican star, took to Facebook, one of the vehicles for promoting opposition, to renounce a bill he had co-sponsored. Senator John Cornyn of Texas, who leads the G.O.P.'s Senate campaign efforts, used Facebook to urge his colleagues to slow the bill down. Senator Jim DeMint, Republican of South Carolina and a Tea Party favorite, announced his opposition on Twitter, which was already boiling over with anti-#SOPA and #PIPA fever.
Then trickle turned to flood — adding Senators Mark Kirk of Illinois and Roy Blunt of Missouri, and Representatives Lee Terry of Nebraska and Ben Quayle of Arizona. At least 10 senators and nearly twice that many House members announced their opposition.
"Thanks for all the calls, e-mails, and tweets. I will be opposing #SOPA and #PIPA," Senator Jeff Merkley, Democrat of Oregon, wrote in a Twitter message. Late Wednesday, Senator Charles E. Grassley of Iowa, the senior Republican on the Senate Judiciary Committee, withdrew his support for a bill he helped write.
The existing bill "needs more due diligence, analysis and substantial changes," he said in a statement.
Few lawmakers even now question the need to combat pirates at Web sites in China, Russia and elsewhere who have offered free American movies, television shows, music and books almost as soon as they are released. Heavyweights like the Walt Disney Company secured the support of senators and representatives before the Web companies were even aware the legislation existed.
"A lot of people are pitching this as Hollywood versus Google. It's so much more than that," said Maura Corbett, spokeswoman for NetCoalition, which represents Google, Amazon.com, Yahoo, eBay and other Web companies. "I would love to say we're so fabulous, we're just that good, but we're not. The Internet responded the way only the Internet could."
For the more traditional media industry, the moment was menacing. Supporters of the legislation accused the Web companies of willfully lying about the legislation's flaws, stirring fear to protect ill-gotten profits from illegal Web sites.
Mr. Dodd said Internet companies might well change Washington, but not necessarily for the better with their ability to spread their message globally, without regulation or fact-checking.
TimesCast | Tech Sector's Clout
January 18, 2012 - As Wikipedia and other websites go dark Wednesday, protesting proposed anti-piracy legislation, a new axis of influence emerges in Washington.
By Emily B. Hager and Kassie Bracken on Publish Date January 18, 2012. . Watch in Times Video »
"It's a new day," he added. "Brace yourselves."
Citing two longtime liberal champions of the First Amendment, Senator Patrick Leahy and Representative John Conyers Jr. of Michigan, Mr. Dodd fumed, "No one can seriously believe Pat Leahy and John Conyers can be backing legislation to block free speech or break the Internet."
For at least four years, Hollywood studios, recording industry and major publishing houses have pressed Congress to act against offshore Web sites that have been giving away U.S. movies, music and books as fast as the artists can make them. Few lawmakers would deny the threat posed by piracy to industries that have long been powerful symbols of American culture and have become engines of the export economy. The Motion Picture Association of America says its industry brings back more export income than aerospace, automobiles or agriculture, and that piracy costs the country as many as 100,000 jobs.
The House response, SOPA, was drafted by a conservative Republican, Representative Lamar Smith of Texas, with the backing of 30 co-sponsors, from Representative Debbie Wasserman Schultz of Florida, the chairwoman of the Democratic National Committee, to mainline Republican Peter King of New York. The Senate's version, written by Mr. Leahy, the Vermont Democrat who is chairman of the Senate Judiciary Committee chairman, attracted 40 co-sponsors from across the political spectrum and cleared his committee unanimously.
Then the Web rose up. Activists said the legislation would censor the Web, force search engines to play policemen for a law they hate and cripple innovation in one of the most vibrant sectors of the American economy.
Mr. Smith, the House Republican author, said opposition Web sites were spreading "fear rather than fact."
"When the opposition is based upon misinformation, I have confidence in the facts and confidence that the facts will ultimately prevail," Mr. Smith said.
Google, Facebook and Twitter have political muscle of their own, with in-house lobbying shops and trade associations just like traditional media's. Facebook has hired the former Clinton White House press secretary Joe Lockhart. Google's Washington operations are headed by Pablo Chavez, a former counsel to Senator John McCain, Republican of Arizona, and a veteran of the Senate Commerce Committee.
And for all the campaign contributions, Washington parties and high-priced lobbyists the old economy could muster, nothing could compare to the tentacles the new economy can reach into Americans' everyday lives through sites like Wikipedia. Aides to Senator Harry Reid, the majority leader, say he will press forward with a vote Tuesday to open debate on the Protect I.P. bill. Negotiators from both parties are scrambling for new language that could assuage the concerns of the Internet community, but expectations are that the bill will now fail to get the 60 votes to move forward — a significant setback.
"The problem for the content industry is they just don't know how to mobilize people," said John P. Feehery, a former House Republican leadership aide who previously worked at the motion picture association. "They have a small group of content makers, a few unions, whereas the Internet world, the social media world especially, can reach people in ways we never dreamed of before."
A version of this article appears in print on January 19, 2012, on Page A1 of the New York edition with the headline: Web Rises Up To Deflect Bills Seen as Threat. Order Reprints| Today's Paper|Subscribe
With Blackouts and Twitter, Web Flexes Its Muscle JAN. 18, 2012
Wikipedia Protest Noticed, but Some Yawn JAN. 18, 2012
Online Piracy and Political Overreach JAN. 18, 2012
Opinion Op-Ed Contributor
SOPA Boycotts and the False Ideals of the Web JAN. 18, 2012
Opinion Room for Debate
What's the Best Way to Protect Against Online Piracy? JAN. 18, 2012
Opinion The Loyal Opposition
Behold the Power of Google JAN. 18, 2012


Summary View down unpaved Eureka Street in Central City, Colorado. The Opera House is in the foreground to the left, and a hitching post and street lamp are in the center. Two men walk down the sidewalk toward the town. Cars are parked on both sides of the street. The Teller House has a sign attached to it that reads, "J.S. Kimball Assay Office." The First National Bank building, the Miller block, and the Hanse block are in the background. The buildings on the opposite side of the street include, the Register block, the Morse block, and the Pharmacy building, which has a sign attached to it that reads, "Drugs."


The death of a man in his Hialeah home Friday left Miami-Dade County Animal Services with more than 50 small to medium sized dogs in need of adoption.
Hialeah Police responded to a call of a man passing away from natural causes Friday morning at a home on the 5900 block of East 6th Avenue. When police arrived, they found the body of a man along with his elderly parents and roughly 50 dogs in the home.
Police said the family loved animals and would pick up stray dogs from the street to try to help them. Hialeah Police said both parents appeared to be suffering from dementia and the dogs were taken by animal control to be put up for adoption.
Both parents were taken to Palmetto General Hospital for further evaluation.
The dogs are at Miami-Dade County Animal Services at 7401 NW 74 Street in Miami. The shelter waived all adoption fees for the dogs. The dogs are available this weekend from 10 a.m. to 4 p.m. on Saturday and Sunday and Monday through Friday from 10 a.m. to 6:30 p.m.


TKD Bra Genie: Are you wearing the wrong size?
At TKD Lingerie we believe that looking and feeling great starts underneath your clothes, and that means wearing the right underwear.
We don't call ourselves D-K Cup Experts for no reason! We are dedicated to matching women with their perfect bra so they can feel fabulous on the inside and the outside. Contrary to what people say, size and fit really do matter!
Astonishingly, up to 80% of women are wearing the wrong size bra! The truth is, standard bra-makers do not always account for a wide range of sizes and shapes. Women with fuller busts often need more support and better engineered bras than those readily available on the market. They also need to be fitted by experts who understand the importance and fit of a well-engineered bra.
We feel like this is an injustice to our fellow females and we've made it our mission to turn this statistic on its head. That's why we're introducing our TKD Bra Genie, to give ladies a chance to find the perfect fit through our easy and convenient personalized fitting service.
Our TKD Bra Genie can tell you your bra size from first sight! How? Simply send in a picture of yourself in a fitted t-shirt, wearing your best fitting unpadded bra and tell us your dress size. It's so easy, you can do so over Whatsapp (+971555115996) or through our Bra Genie fitting page (http://www.tkdlingerie.com/bra-genie/ ). An expert will be in touch with you before you know it to advise on your perfect fit.
A great bra does much more than just provide everyday comfort! It can improve posture, silhouette and even make you look a dress size smaller. With TKD Lingerie's amazing selection of bras, you can find styles that do all of these and look pretty too!
What are you waiting for? Feel the benefits of our TKD Bra Genie and experience the wonder of what great fitting underwear can do!


TORFAEN council must be "cautious" about any future use of a controversial weedkiller, a meeting has heard.
Glyphosate, a powerful herbicide commonly used across the world, has proved controversial after being link with cancer during a landmark US court case.
Several councils in Wales are now reviewing their use of the chemical despite it being approved for use until 2022 by the European Union.
In Torfaen, contractors spray glyphosate on pavements twice a year while in-house staff tackle Japanese knotweed on council land by spraying or injecting the substance into the root.
The cleaner overview and scrutiny committee heard arguments for and against its continued use on Wednesday.
Carole Jacob, of Torfaen Friends of the Earth, said the council should commit to phasing out glyphosate and engage and research alternative methods.
"We are of the belief that glyphosate is not environmentally friendly and not good for human health," said Mrs Jacob.
Mrs Jacob added that she had been "highly incensed" by supermarkets stocking glyphosate products in food stores.
But co-opted committee member Alun Williams remained sceptical about the evidence, saying that there had been no outcry prior to the Monsanto court case.
"This case was based on the decision of 12 laypeople at one time, and that's being appealed," he added.
Councillors also heard from Alan Abel and Chris Phillips, directors of Complete Weed Control which is contracted by Torfaen council.
Mr Abel, who has worked with glyphosate since it was first introduced in the 1970s, said the minimum amount is used during each application.
"We've cut the amount of glyphosate used in south Wales by up to 80 per cent," he said.
Mr Phillips described glyphosate, which is heavily diluted for use in public areas, as the "strongest tool in our armament".
Acetic acid, or horticultural vinegar, had been trialled as an alternative but was later withdrawn in Bristol because it was "expensive and smelt like a fish and chip shop", according to Mr Abel.
The meeting heard from council officers that glyphosate was the most efficient way of getting rid of Japanese knotweed.
Councillor Norma Parish hoped that the council could eventually stop using glyphosate, saying: "I would stop the spraying of pavements immediately.
The committee recommended that the current provisions are maintained until further research is released.
They also asked for the public to be better informed of when and where sprays are being carried out in the borough.


Horse lovers dream! Post & Beam Log sided expanded cape with detached garage / barn attached with 2 Horse Stalls, water, electricity and an additional 36 x 40 out Building for your Tack shop or other home business set up all located on 95 acres of combined farmland, trails and certified NH. tree farm. Bring your Family , horses and other livestock / farm animals to enjoy the large pastures , stone walls, private trails for walking , horseback riding, snowmobiling, and ATV adventures. Enjoy camping or hunting next to your own 5 acre +/- pond or gardening on the acres of tilled rich soil ripe for vegetables and perennials. Make Autumn Harvest Farm your family Estate or Retreat from the hustle and bustle of the city life! Close proximity to skiing, hiking , Dartmouth College and the New London Lake Sunapee region. This beautiful sprawling cape features single level living with a large Master Bedroom on the main floor, oversized kitchen area with beautiful cherry cabinets, gas cooktop and 3 ovens for the serious baker. Large living room with stone hearth fireplace / woodstove and laundry room all on the main level. Upstairs currently used as a Bed & Breakfast center with 3 guest rooms a breakfast nook and The Jacuzzi suite complete with Jet tub, bedroom and studio sitting area. This property is a unique NH. treasure that must be scene to appreciate all that it has to offer! See MLS Property & Listing Details & 34 images.
Sitting back off the road this contemporary home has 11 acres of privacy. Huge metal garage, and an additional two more sheds for storing all your toys and supplies. Being exceptionally well insulated, this 3 bedroom home is energy efficient, with a wood stove, pellet stove and Rinnai propane heater. Lots of light beaming in the large windows in the open concept living room with cathedral ceilings. Large enclosed screened porch for your quiet, bug free enjoyment of the outdoors. See MLS Property & Listing Details & 20 images.
With Ragged Mountain Ski Area, the Northern Rail Trail for snowmobiling and recreation, area lakes and ponds nearby with public access for swimming and fishing nearby this 1840 Colonial could make a wonderful "turn key" second home to an extended family. "The Maples" constructed in 1840 has been loved and updated tastefully in recent years. You can enjoy the charm of the original features of wide pine floors, exposed beams, built ins, and original staircase. There is a 13x24 eat in kitchen or a more formal dining room with french doors to the rear yard. 4 bedrooms that will fit any size bed. 2 full baths, a study, a 24x25 family room with wide pine floors, built in book case, exposed beams, & 2 closets. Living Room with hearth & wood stove, and built in cabinets. Replacement windows, and updated systems in recent years including a new hot water tank in 2018. 5.38 surveyed acres with fields, Woodlot, 3 ponds, a trestle bridge, blueberries, garden spot, screen house to enjoy your summer nights, storage shed, and bunk house. With rolling fields the land offers rich soils for farming, gardening, or farm animals. See MLS Property & Listing Details & 40 images.
Here you have two houses on 10.5 acres which represents income potential towards payment of your mortgage!!! The main house is approximately 1/4 of a mile down the driveway, while the second house is at the front of the property close to the road. The main house has 2 bedrooms and 1 bath with a state approved septic system installed rated for 3 bedrooms. The second home is 2 bedrooms and 1 bath and has a state approved septic system installed rated for 2 bedrooms. Close drive to Grafton Pond. Come and check out all the possibilities this property has to offer. See MLS Property & Listing Details & 17 images.
This 4 bedroom country cape is on 6.4 acres+/- with rolling fields, woodlands, streams, and frontage on 2 town roads and US Route 4. The kitchen and bathrooms have been updated, there are wide pine and hardwood floors, exposed beams in the living room, vinyl siding, replacement windows, and a FHA oil and separate pellet furnace. 4 bedrooms, 3 baths, mudroom to the new 22x11 deck. New whole house Guardian Generator installed in September 2018, kitchen appliances, and washer and dryer are included. This home is wheel chair accessible and has a handicap shower on the first floor. Detached 1 car garage with side storage, paved driveway and walk. New 4 bedroom state approved septic system installed in December 2018. Come take a look with us! See MLS Property & Listing Details & 34 images.
$230,000 | $180 per sq.ft. GRAFTON NH HOME FOR SALE New Listing!
Well maintained, built in 2006 this home is move in ready, economical heating costs, vinyl siding, make this ranch with 3 bedrooms, 2.5 baths, one that you should consider for your purchase this year. Open area dining/living with cathedral ceilings, hardwood vanities & kitchen cabinets, kitchen with built in microwave, refrigerator, dishwasher, and gas range. The full walk out basement has been finished in recent years into a family room, gym room for work outs, and 1/2 bath. There is a workshop and unfinished storage/utility room. 8x40 deck for summer fun! 30 minutes to DHMC, minutes to Ragged Mtn for skiing, Area ponds and lakes with public access. Ready to move into & put your feet up!! See MLS Property & Listing Details & 40 images.
Early 1800's Post and Beam Federal Style home located on the former stage coach route from Concord to Montreal. A spectacular site with rolling fields, maple trees, stone walls, and 180 degree view of fields and mountains. You will enjoy the cool breezes in the summer and the quiet! Miles of old road for horse back riding, hiking, or snow snowmobiling. This grand home has the original hand hewn beams, gun stock corners, large sun filled rooms, and recent updates of new roof shingles, siding, and a FHA heating system. Nice floor plan with first floor bedroom, wood floors, stairs to the full attic. There is an automatic generator system that powers the entire house, 2 bay attached garage with workshop & mudroom. See MLS Property & Listing Details & 37 images.
Custom Built 2 bedroom, 1 bath ranch with a full unfinished walk out basement for future expansion with windows and a walk out door. FHA/gas heat, central air, or heat with the woodstove in the living room. Nice open concept living and kitchen area with wood beamed ceilings, Marvin windows, R50 in the ceiling, vinyl siding for easy maintenance, 20x36 detached, insulated, and heated garage with overhead storage. Private setting with potential for subdivision, walk road to the height of land with view of Ragged Mountain. 10 acres+/- of hardwood forest, beautiful stone walls, and a hardpack driveway. See MLS Property & Listing Details & 40 images.
This lovely-turn key home boasts a spacious kitchen and dining room with ceramic tile flooring. Open concept living room with hardwood floors, and wool carpets in the bedrooms. Anderson windows, and energy saving radiant heat and a new Harmon pellet stove to keep you extra warm on those cold nights. The porch provides a lovely quite place to read a book or just relax while enjoying the babbling brook nearby. The dry walkout basement is ready for expansion with plenty of natural light. Also a large two car garage with extra storage above. This home in includes a great patio and perennial gardens for those with a green thumb. If you love the outdoors you'll appreciate the shared access to the private waterfront at Kilton Pond for recreational opportunities year round. This surprisingly large home must be seen to be appreciated. Book your appointment to view this home today. See MLS Property & Listing Details & 40 images.