text:
\section{Introduction}

In the present paper we are interested with the existence of solutions for the following class of equations
\begin{equation}\label{i1}
-\Delta_p u + a(x)|u|^{p-2}u + \lambda \Psi |u|^{p-2}u = 0 \textrm{ \ in \ } \mathbb{R}^N,
\end{equation}
where $ a:\mathbb{R}^{N} \rightarrow \mathbb{R} $ is a continuous non-negative function, $ \lambda >0 $, $ N \geq 3 $ and $ \Psi : \mathbb{R}^{N} \rightarrow \mathbb{R} $ stands for an internal potential for a nonlocal self-interaction of $ u $.

Consider $ u $ a radial function, that is, $u(x)=u(r),\ r=|x|,$ $ r>0 .$  Then, in similarity with the case $ p=2 $, we have a \textit{``fundamental solution for the $ p $-laplacian''} equation, $ -\Delta_p u =0 $, given by
$$
\Gamma_p (x) = \left\{ \begin{array}{rcl}
\dfrac{1}{(p-N)\omega_{N-1}}|x|^{1-\frac{N-1}{p-1}} &\textrm{ if \ } &N\neq p ,\\
\dfrac{1}{\omega_{N-1}}\ln |x|& \textrm{ if \ }& N= p ,
\end{array} \right.
$$
where $ \omega_{N-1} $ is the $ (N-1)$-dimensional measure of the $ (N-1)$-sphere.

In this sense, we can consider equation (\ref{i1}) with $ \Psi = \Gamma_p \ast |u|^p $, that is
\begin{equation}
-\Delta_p u + a(x)|u|^{p-2}u + \lambda (\Gamma_p \ast |u|^p) |u|^{p-2}u = 0 \textrm{ \ in \ } \mathbb{R}^N .
\end{equation}
In this work we will focus on finding solutions for the p-planar case, $ N=p $, and with a nonlinearity of Moser-Trudinger type,
\begin{equation} \label{P}
-\Delta_N u + a(x)|u|^{N-2}u + \lambda (\ln|\cdot|\ast |u|^{N})|u|^{N-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N,
\end{equation}
with $ a \equiv 1 $, $ \lambda = \omega_{N-1} $, $ N \geq 3 $ and $f: \mathbb{R} \rightarrow [0, \infty) $ a continuous function that behaves like $ \exp(\alpha |u|^{\frac{N}{N-1}}) $ at infinity, with primitive $ F(s)=\int\limits_{0}^{s}f(t)dt $.

In the sequence, we make a quick overview of literature. We start mentioning that Choquard equations are well-known for its great range of applications in science, specially in physics and mechanics.  The case $ N=3 $ for Laplacian operator has been extensively studied, due to its relevance in physics. Although this equation is called ``Choquard equation'', its seminal paper is due to Fröhlich and Pekar in \cite{[12] , [11] , [22]}, where the authors describe the quantum mechanics of a polaron at rest, in the particular case, when $ V(x)\equiv a > 0 $ and $ \gamma > 0 $. Then, it was considered by Choquard in 1976, while studying an electron trapped in its hole. We highlight that the local nonlinear terms on the right side of these equations, such as $ f(u)= b|u|^{p-2}u $, for $ b\in \mathbb{R} $ and $ p>2 $, stands, usually, in Schrödinger equations as a modelling of the interaction among particles.

Since the mentioned seminal works, several variations of Choquard equations have been considered as a way to model different phenomena. As an additional example, we cite \cite{[18]} where Penrose has derived a Choquard equation while discussing about the self gravitational collapse of a quantum-mechanical system. Moreover, many results have been derived recently about the existence and regularity of solutions, including more general convolution potentials, such as in the evolution equation $ i\partial_t \phi = \Delta \phi +(V \ast |\phi|^2)\phi,  $ which models the interaction of a large system of nonrelativistic bosonic atoms and molecules.

We also refer the reader to \cite{[20] , [16]}, in which the authors treat variations of Choquard equation with $ N\geq 3 $ and $ p=2 $.

When dealing with specifically logarithmic forms of Choquard equations, one has scantier literature. It took some time for scientists to figure out how to treat this case, because, the presence of the logarithmic function rises a bunch of difficulties. Specially while dealing with Cerami sequences. In this sense, we can cite some seminal and recent works \cite{[6], [cjj], [10], [4]} where the authors have proved the existence and multiplicity of solutions for Laplacian equations with polynomial nonlinearity. Moreover, in \cite{[alves]}, the authors have dealt with equation (\ref{P}) considering a nonlinearity with exponential critical growth and Laplacian operator. Finally, in \cite{[boer2]}, the authors prove existence and multiplicity results for the fractional $ p-$Laplacian operator with exponential growth nonlinearity. In problems with such critical behaviour, some additional difficulties arises while dealing with Cerami sequences and that is why authors need some stronger conditions over the nonlinearity.

On the other hand, problems involving the $N$-Laplacian operator are important in many fields of sciences. Since they can accurately describe the behaviour of electric, gravitational and fluid potentials, they mostly appear in the fields of electromagnetism, astronomy and fluid dynamics. Moreover, the study of ground state solutions for the $N$-Laplacian equation is crucial in the research of evolutions equations which appear in non-Newton fluids, turbulent flows in porous media and other contexts.

We refer the reader to \cite{[medeiros], [lam], [zhang]} and the references therein, in which the authors work or refer to work where many  different  cases of equations involving the $N$-Laplacian operator are studied, such as  bounded  domains  and  unbounded  domains,  different behaviour of nonlinearities, different types of boundary conditions, and so on. Particularly, many works focussed on the study of subcritical and critical growth for the involved nonlinearities which allows us to treat the problem variationally.

Our work intend to extend or complement the results found in the above papers. The main features of the class of problems considered in this paper, are that it is defined in the whole $\mathbb{R}^{N}$, it has a convolution term with the logarithmic function, which changes sign and in unbounded from above and bellow, involves an exponential critical growth and the nonlinear operator $N$-Laplacian. Such combination of element drives to new possibilities of applications.

Now we present the conditions imposed over the nonlinearity $ f $ in order to obtain the desired results. Such hypothesis are usual in works involving a Moser-Trudinger inequality and the $ N $-Laplacian operator, as one can see in \cite{[doO], [zhang], [lam]}.

In order to simplify some calculations and make the notation more concise, we will write
$$ R(\alpha , s) = \exp(\alpha |s|^{\frac{N}{N-1}}) - S_{N-2}(\alpha , s) = \sum\limits_{N-1}^{+\infty} \dfrac{\alpha^k}{k!} |s|^{\frac{N}{N-1}k} , $$
where $S_{N-2}(\alpha_0 , s) = \sum\limits_{k=0}^{N-2}\dfrac{\alpha_{0}^{k}}{k!}|s|^{\frac{N}{N-1}k}$.

We recall that a function $ h $ is said to have \textit{subcritical} exponential growth at $ +\infty $, if
$$
\lim\limits_{s\rightarrow + \infty}\dfrac{h(s)}{R(\alpha , s)} = 0 \textrm{ \ , for all \ } \alpha >0 ,
$$
and we say that $ h $ has $ \alpha_0 $-\textit{critical} exponential growth at $ +\infty $, if
$$
\lim\limits_{s\rightarrow + \infty}\dfrac{h(s)}{R(\alpha , s)} = \left\{ \begin{array}{ll}
0, \ \ \ \forall \ \alpha > \alpha_{0} \\
+\infty , \ \ \ \forall \ \alpha < \alpha_0
\end{array} \right. .
$$
In this sense, we assume that $f$ has critical growth. Moreover, motivated by Lemma \ref{l1}, we assume the following growth condition
$$ f\in C(\mathbb{R} , \mathbb{R}), f(0)=0 \mbox{ and has critical exponential growth.} \leqno{(f_1)} $$
$$ \lim\limits_{s\rightarrow 0^{+}} \dfrac{f(s)}{|s|^{N-1}}=0.  \leqno{(f_2)}$$

Before giving the next condition, we point out that, following the idea introduced by Stubbe \cite{[21]}, we will consider the slightly smaller Hilbert space
\begin{equation} \label{X}
X = \left\{ u\in  W^{1,N}(\mathbb{R}^{N})   ; \displaystyle\int\limits_{\mathbb{R}^N} \ln(1+|x|)|u(x)|^N dx < \infty \right\} \subset  W^{1,N}(\mathbb{R}^{N})  ,
\end{equation}
endowed with the norm
$$
||u||_{X}^{N}=||u||^{N}+||u||_{\ast}^{N} \mbox{ \ \ , \ \ where \ \ } ||u||_{\ast}^{N}=\displaystyle\int\limits_{\mathbb{R}^N} \ln(1+|x|)|u(x)|^{N} dx ,
$$
in which the portion of the associated functional involving the $ \ln |\cdot| $ function will be finite.

Next, we need some kind of Ambrosetti-Rabinowitz condition to guarantee that Cerami sequences in $ X $ are bounded in $  W^{1,N}(\mathbb{R}^{N})  $. The reader can see that this condition is a bit stronger that some usually found in the literature. It happens because two main causes, which are the integral term involving the logarithmic and the exponential behaviour of $ f $ at infinity. Once we need a bit more information about $ ||\nabla \cdot||_N $ to get the boundedness of the integral involving the exponential term, we cannot even adapt the conditions of \cite{[6], [cjj], [10]}, for example. In this sense, we ask for

$$ \mbox{ there exists }\ \theta \geq  2N \ \mbox{ such that}\   f(s)s \geq \theta F(s) > 0, \ \mbox{for all }\  s\in \mathbb{R} \setminus \{0\}. \leqno{(f_3)}$$

However, while the above condition exclude the necessity of dealing with the integral involving the exponential term, it is not always possible to argue that way. So, based on works such as \cite{[9]}, we insert a condition that makes possible to get a boundedness for the integral involving the exponential term and add some other geometric properties for the associated functional.

$$ \mbox{ there is}\  q>2N \ \mbox{ and}\  C_q> \dfrac{S_{q}^{q}[2(q-N)]^{}\frac{q-N}{N}}{\rho_{0}^{q-N}q^{\frac{q}{N}}} \ \mbox{ such that}\  F(s) \geq C_q |s|^q , \ \mbox{for all}\  s\in \mathbb{R} , \leqno{(f_4)}$$
where $ S_q $ is a constant obtained from the Sobolev embeddings and $ \rho_0 $ is a suitable value such that the exponential term can be controlled, defined in Lemma \ref{l10}.

Our study is based on variational techniques. In this sense, we will give some necessary informations about the problem (\ref{P}). One can see that the associated functional $ I:  W^{1,N}(\mathbb{R}^{N})  \rightarrow \mathbb{R} \cup \{ \infty\} $ is given by
$$
I(u)=\dfrac{1}{N}\displaystyle\int\limits_{\mathbb{R}^N} |\nabla u|^N + |u|^N dx + \dfrac{1}{2N} \displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln(1+|x-y|) |u(x)|^N |u(y)|^N dx dy - \displaystyle\int\limits_{\mathbb{R}^N} F(u) dx .
$$
As we mentioned earlier, $ I $ is not well-defined in $  W^{1,N}(\mathbb{R}^{N})  $. But, it is possible to see that, for any $ u\in X $, from Moser-Trudinger Lemma \ref{l1}, $ I(u)< +\infty $. Moreover, by standard computations one can see that the Gateaux derivative of $ I $ is given by
\begin{small}
$$
I'(u)v = \displaystyle\int\limits_{\mathbb{R}^N} [|\nabla u|^{N-2}\nabla u \nabla v + |u|^{N-2}uv] dx + \displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln(1+|x-y|) |u(x)|^{N} |u(y)|^{N-2}u(y)v(y) dx dy - \displaystyle\int\limits_{\mathbb{R}^N} f(u)v dx
$$
\end{small}
and that $ I\in C^1(X, \mathbb{R}) $. Also, $ I $ is clearly invariant under $ \mathbb{Z}^N $ translations. Hence, noting that any critical point of $ I $ is a weak solution for (\ref{P}), we will be concerned in finding critical points for $ I $.

Provided the necessary background, we present our main result concerning the existence of solutions for (\ref{P}).

\begin{theorem}\label{t1}
Assume $ (f_1)-(f_4) $, $ q>2N $ and that $ C_q>0 $ is sufficiently large. Then,
\begin{itemize}
\item[(i)] Equation (\ref{P}) has a solution $ u\in X\setminus \{0\} $ with $ I(u)=c_{mp} $, where
\begin{equation}\label{eq9}
c_{mp}= \inf\limits_{\gamma \in \Gamma} \max\limits_{t\in [0, 1]} I(\gamma(t)) ,
\end{equation}
with $ \Gamma = \{ \gamma\in C([0, 1], X) \ ; \ \gamma(0)=0 \ , \ I(\gamma(t))< 0 \} $.

\item[(ii)] Equation (\ref{P}) has a non-trivial ground state solution, i.e there exists $ u\in X\setminus \{0\} $ such that $$ I(u)=c_g = \inf\{I(v) \ ; \ v\in X\setminus\{0\} \mbox{ \ is a solution of (\ref{P}}) \} . $$
\end{itemize}
\end{theorem}

\begin{corollary}\label{c400}
We have that $c_g > 0$. Particularly, problem \eqref{P} has only positive energy level non-trivial solutions.
\end{corollary}

For the second main result, we are concerned with multiplicity of solutions. However, to obtain guarantee that $ I $ has the necessary geometry, we need to ask for $ (f_5) $ and replace the condition $ (f_1) $ by the condition below.
$$ f\in C(\mathbb{R} , \mathbb{R}), \ f \mbox{ \ is odd and has subcritical exponential growth.} \leqno{(f_1 ')}$$

$$ \mbox{the function \ } t\mapsto \dfrac{f(t)}{t^{2N-1}} \mbox{ \ is increasing in \ } (0, + \infty) . \leqno{(f_5)}$$
From $ (f_5) $, it follows that $ \frac{f(t)}{t^{2N-1}} $ is decreasing in $ (-\infty , 0) $.

Moreover, since we can control the exponent using $ \alpha >0 $, we can change condition $ (f_4) $ by a more general condition, that is
$$\mbox{ there  exists } q>2N \mbox{ \ and \ } M_1 > 0 \ \mbox{ such that}\  F(t) \geq M_1 |t|^q \ , \ \forall \ t\in \mathbb{R} , \leqno{(f_4 ')}$$

\begin{theorem}\label{t2}
Suppose $ (f_1 '), (f_2), (f_3), (f_4 '), (f_5) $. Then, problem (\ref{P}) has infinitely many solutions.
\end{theorem}

Once the functional $ I $ has the same geometry as the functional associated to the problem in \cite{[boer2]}, we will not repeat the details of the proofs. The reader can found it full detailed in \cite[Section 5]{[boer2]}. One should observe that the arguments are mainly based in the geometry of $ I $ independent of the specific form of the Laplacian type operator involved.

Throughout this paper, we will use the following notations: $  L^{s}(\mathbb{R}^N) $ denotes the usual Lebesgue space with norm $ ||\cdot ||_s $ \ ; \ $ X' $ denotes the dual space of $ X $ \ ; \ $ B_r(x) $ is the ball centred in $ x $ with radius $ r>0 $ \ ;  \ $ C, C_2, ... $ will denote different positive constants whose exact values are not essential to the exposition of arguments.

The paper is organized as follows: in section 2 we present some technical and essential results, some of them already derived in previous works and whose application to our problem is immediate. Finally, section 3 consists in the proof of a key proposition and our main result.

\section{Framework and Preliminary Results}

In this section, we will provide some extra framework informations and present very useful technical lemmas. The results that already possess fully proofs in the other works, we will only make a fast recall, in order to focus on our new results.

Inspired by \cite{[6]}, we start defining three auxiliary functionals, $V_1:  W^{1,N}(\mathbb{R}^{N})  \rightarrow [0, \infty],$ $V_2: L^{\frac{2N^2}{2N-1}}(\mathbb{R}^{N}) \rightarrow [0, \infty) $ and $V_0:  W^{1,N}(\mathbb{R}^{N})  \rightarrow \mathbb{R} \cup \{\infty\},$ given by
\begin{equation*}
u \mapsto V_1(u)=\displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln(1+|x-y|)|u(x)|^N |u(y)|^N dx dy ,
\end{equation*}
\begin{equation*}
u \mapsto V_2(u)=\displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln\left(1+\dfrac{1}{|x-y|}\right)|u(x)|^N |u(y)|^N dx dy ,
\end{equation*}
\begin{equation*}
u \mapsto V_0(u)=V_1(u, v)-V_2(u, v)=\displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln(|x-y|)|u(x)|^N |u(y)|^N dx dy.
\end{equation*}
These definitions are understood to being over measurable functions $u: \mathbb{R}^{N} \rightarrow \mathbb{R} $, such that the integrals are defined in the Lebesgue sense. We also recall that, for $ r > 0 $, $ \ln(1+r) \leq r $ and
\begin{equation}\label{eq12}
\ln(1+|x-y|) \leq \ln(1+|x|)+\ln(1+|y|)  \ , \ \forall \ x, y \in \mathbb{R}^{N} .
\end{equation}

The proofs of the next two technical lemmas follow some standard arguments (see for example \cite{[6], [boer2]}).

\begin{proposition} \label{p3}
\begin{itemize}
\item[(i)] The space $ X $ is compactly embedded in $  L^{s}(\mathbb{R}^N) $, for all $ s\in [N, +\infty) $.

\item[(ii)] The functionals $ V_0, V_1, V_2 $ and $ I $ are of class $ C^1 $ on $ X $. Moreover, for any $ u, v\in X $,
$$
V_{1}'(u)(v)=2N\displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln(1+|x-y|)|u(x)|^N |u(y)|^{N-2}u(y)v(y) dxdy ,
$$
and for any $ u, v \in L^{\frac{2N^2}{2N-1}}(\RN) $,
$$
V_{2}'(u)(v)=2N\displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln\left(1+\dfrac{1}{|x-y|}\right)|u(x)|^N |u(y)|^{N-2}u(y)v(y) dxdy .
$$

\item[(iii)] Functional $ V_2 $ is continuous (in fact continuously differentiable) on $L^{\frac{2N^2}{2N-1}}(\RN)$ .

\item[(iv)] Let $ (u_{n})\subset X $ such that $ u_{n} \rightharpoonup u $ in $ X $. Then,
$$
\lim\limits_{n\rightarrow + \infty} \displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln(1+|x-y|)|u_{n}(x)|^N |u(y)|^{N-2} u(y) (u_{n}(y) - u(y)) dx dy = 0 .
$$
\end{itemize}
\end{proposition}

From a corollary of Ergorov's Theorem, we can state the next proposition, which proof can be done similarly as \cite[Proposition 3.1]{[boer2]}, with $ p=N $.

\begin{proposition}\label{p2}
Let $ u\in  L^{N}(\mathbb{R}^N)  \setminus \{0\} $, $ (u_{n})\subset  L^{N}(\mathbb{R}^N)  $ such that $ u_{n}(x)\rightarrow u(x) $ a.e in $ \mathbb{R}^{N} $ and $ (v_{n})\subset  L^{N}(\mathbb{R}^N)  $ bounded. Set
$$
\omega_n = \displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln(1+|x-y|)|u_{n}(x)|^N |v_{n}(y)|^N dx dy .
$$
If $\sup\limits_{n} \omega_n < + \infty $, then $ ||v_{n}||_\ast $ is bounded. Moreover, if $ \omega_n \rightarrow 0 $ and $ ||v_{n}||_N \rightarrow 0 $, then $ ||v_{n}||_\ast \rightarrow 0 $.
\end{proposition}
\begin{proof}
From the Ergorov's corollary, there exists $ R\in \mathbb{N} $, $ \delta > 0 $, $ n_0 \in \mathbb{N} $ and $ A\subset B_R $, such that $ A $ is measurable, $ \mu(A)>0 $ and $ u_{n}(x) > \delta $, for all $ x\in A $ and for all $ n \geq n_0 $. Then, by some calculation, one can obtain
\begin{equation}
\omega_n \geq \dfrac{\delta^N \mu(A)}{2}  (||v||_{\ast}^{N}-\ln(1+2R)||v_{n}||_{N}^{N}),
\end{equation}
from which the result follows.
\end{proof}

Let $ u\in X $. By Hardy-Littlewood-Sobolev Inequality (HLS) \cite{[15]} with $ \alpha, \beta =0  $, $ \lambda = 1 $, $ g(x)=|u(x)|^N $, $ f(y)=|u(y)|^N $ and $ \frac{1}{q}+\frac{1}{t}+\frac{1}{N}=2 $, we have
\begin{equation} \label{v2}
|V_2(u)|\leq K_0||u||_{\frac{2N^2}{2N-1}}^{2N} \ , \ \ \forall \ u\in L^{\frac{2N^2}{2N-1}}(\mathbb{R}^{N}) ,
\end{equation}
where $ K_0 $ is the HLS constant. Particularly, $ V_2 $ takes finite values over $ L^{\frac{2N^2}{2N-1}}(\mathbb{R}^{N})\subset  W^{1,N}(\mathbb{R}^{N})  $.

\begin{observation}\label{obs8} From \eqref{eq12},
$$\displaystyle\int\limits_{\mathbb{R}^N} \displaystyle\int\limits_{\mathbb{R}^N} \ln(1+|x-y|)|u(x)|^N |v(y)|^N dxdy \leq ||u||_{\ast}^{N}||v||_{N}^{N}+ ||v||_{\ast}^{N}||u||_{N}^{N}.$$
As a consequence, $ V_1(u)\leq 2||u||_{\ast}^{N}||u||_{N}^{N} $, for all $u\in  W^{1,N}(\mathbb{R}^{N}) $.
\end{observation}

Before discuss further the exponential behaviour, we would like to highlight some important inequalities obtained from our main assumptions on $ f $.

\begin{observation}\label{obs1}
Given $ \varepsilon >0 $, $ q>N $ and $ \alpha > \alpha_0 $, from $ (f_2) $ and $ (f_1) $, for all $ u\in  W^{1,N}(\mathbb{R}^{N})  $, there exists a constant $ b_1 >0 $ such that
\begin{equation}\label{eq1}
|F(u)|\leq \dfrac{\varepsilon}{N}|u|^N + b_1 |u|^q R(\alpha ,u) .
\end{equation}
Similarly, there exists constants $ b_2, b_3 >0 $ satisfying
\begin{equation}\label{eq14}
|f(u)| \leq \varepsilon |u|^{N-1} + b_2 |u|^{q-1}R(\alpha , u)
\end{equation}
\end{observation}

Next, we provide some extremely important lemmas concerning the exponential growth. Lemma \ref{l1} is the well-known Moser-Trudinger Lemma for $ N $-Laplacian. Lemma \ref{l33} provide us with an essential inequality for potentials of $ R(\alpha , u) $ and then we have some immediate corollaries derived from both.

\begin{lemma}\label{l1}
(Moser-Trudinger Lemma \cite{[doO], [17]}) Let $ N\geq 2 $, $ \alpha >0 $ and $ u\in  W^{1,N}(\mathbb{R}^{N})  $. Then,
$$
\displaystyle\int\limits_{\mathbb{R}^N} [ \exp(\alpha |u|^{\frac{N}{N-1}}) - S_{N-2}(\alpha , u)] dx < \infty ,
$$
where $ S_{N-2}(\alpha, u) = \sum\limits_{k=0}^{N-2}\dfrac{\alpha^{k}}{k!}|u|^{\frac{N}{N-1}k} $. Moreover, if $ ||\nabla u||_{N}^{N} \leq 1 $, $ ||u||_N \leq M < \infty $ and $ \alpha < \alpha_N = N\omega_{N-1}^{\frac{1}{N-1}} $, where $ \omega_{N-1} $ is the $ (N-1) $-dimensional measure of $ (N-1) $-sphere, then there exists a constant $ C_0 = C(\alpha , N , M) $ such that
$$
\displaystyle\int\limits_{\mathbb{R}^N} [ \exp(\alpha |u|^{\frac{N}{N-1}}) - S_{N-2}(\alpha , u)] dx \leq C(\alpha , N , M) = C_0 .
$$
\end{lemma}

\begin{lemma}\label{l33}
(\cite[Lemma 2.3]{[w6]}) Let $ \alpha > 0 $ and $ r>1 $. Then, for every $ \beta > r $, there exists a constant $ C_\beta = C(\beta) > 0 $ such that
$$
(\exp(\alpha |t|^{p'}) - S_{N -2}(\alpha ,t))^r \leq C_\beta (\exp(\beta \alpha |t|^{p'} - S_{N -2}(\beta \alpha , t)) ,
$$
with $ \frac{1}{p}+\frac{1}{p'}=1 $.
\end{lemma}

\begin{corollary}\label{obs1}
Let $ \alpha>0 $. Then, $ R(\alpha , u)^l \in L^{1}(\mathbb{R}^{N}) $, for all $ u\in  W^{1,N}(\mathbb{R}^{N})  $ and $ l \geq 1 $.
\end{corollary}

\begin{corollary}\label{l11}
Let $ u\in  W^{1,N}(\mathbb{R}^{N})  $, $ r>N $, $ l\geq 1 $, $ \beta > 0 $ and $ ||u||\leq M $, for $ M>0 $ sufficiently small. Then, there exists a constant $ K_1 = K_1(\beta , N, M, l, s) > 0 $ such that
$$
\displaystyle\int\limits_{\mathbb{R}^N} |u|^r R(\beta , u)^l dx \leq K_1 ||u||_{t_0}^{r} ,
$$
for some $ t_0 > N $. Moreover, there exists a constant $ K_2 = K_2(\beta , N, M, l, s) > 0 $ such that
$$
\displaystyle\int\limits_{\mathbb{R}^N} |u|^r R(\beta , u)^l dx \leq K_2 ||u||_{X}^{r} .
$$
\end{corollary}

Analyzing Corollary \ref{l11}, one can see that it is possible to obtain the estimatives either controlling the norm $ ||\cdot|| $, as we presented, or controlling the exponent $ \beta $ instead. Both forms are equally important inside of the work. In the next two lemmas we guarantee that $ I $ possess the Mountain-Pass geometry. Their proof are usual and will be omitted (see \cite[Lemmas 3.4 and 3.5]{[boer2]}).

\begin{lemma}\label{l6}
There exists $ \rho > 0 $ such that
\begin{equation}\label{eq6}
m_\beta = \inf \{ I(u) \ ; \ u\in X \ , \ ||u||=\beta \} \mbox{ , for all \ } 0 < \beta \leq \rho
\end{equation}
and
\begin{equation}\label{eq7}
n_\beta = \inf \{ I'(u)(u) \ ; \ u\in X \ , \ ||u||=\beta \} \mbox{ , for all \ } 0 < \beta \leq \rho .
\end{equation}
\end{lemma}
\begin{proof}
Let $ u\in X\setminus \{0\} $, $ \varepsilon > 0 $ small and $ q> N $, then
$$
I(u) \geq \dfrac{||u||^N}{N}[1-\varepsilon - C_1||u||^N - C_2||u||^{q-N}] \mbox{ \ \ \ and \ \ \ } I'(u)(u)\geq ||u||^N (1-\varepsilon - C_3||u||^N - C_4 ||u||^{q-N}).
$$
\end{proof}

\begin{lemma}\label{l7}
Let $ u\in X\setminus \{0\} $, $ t>0 $ and $ q> 2N $. Then,
$$
\lim\limits_{t\rightarrow 0} I(tu) = 0 \ \ , \ \ \sup\limits_{t>0} I(tu) < +\infty \ \ \mbox{and} \ \ I(tu)\rightarrow + \infty \ , \ \mbox{as} \ t \rightarrow + \infty .
$$
\end{lemma}
\begin{proof}
Let $ u\in X\setminus \{0\} $, $ t>0 $ and $ q> 2N $. Then, we observe that
$$
I(tu) \leq \dfrac{t^N}{N}||u||^N+\dfrac{t^{2N}}{2N}V_0(u)-C_q t^{q}||u||_{q}^{q}.
$$
\end{proof}

Since $ I $ has the mountain pass geometry, the mountain pass level defined in (\ref{eq9}) is well-defined and satisfies $ 0< m_\rho \leq c_{mp} < + \infty $. Moreover, there exists a Cerami sequence in the level $ c_{mp} $.

In this sense, consider a sequence $ (u_{n})\subset X $ satisfying
\begin{equation}\label{eq8}
0 < d = \sup\limits_{n\in \mathbb{N}} I(u_{n}) < + \infty \mbox{ \ \ \ and \ \ \ } ||I'(u_{n})||_{X'}(1+||u_{n}||_X) \rightarrow 0 \mbox{ \ as \ } n \rightarrow + \infty .
\end{equation}
Then, by a direct calculation, we get the following lemma.

\begin{lemma}\label{l8}
Let $ (u_{n}) \subset X $ satisfying (\ref{eq8}). Then, $ (u_{n}) $ is bounded in $  W^{1,N}(\mathbb{R}^{N})  $.
\end{lemma}
\begin{proof}
From (\ref{eq8}) and $ (f_3) $, we have
$$
d+ o(1)  \geq I(u_{n}) - \dfrac{1}{2N} I'(u_{n})(u_{n})\geq \dfrac{1}{2N}||u_{n}||^N +  \left(\dfrac{\theta}{2N} - 1 \right) \displaystyle\int\limits_{\mathbb{R}^N} F(u_{n}) dx \geq \dfrac{1}{2N}||u_{n}||^N .
$$
\end{proof}

From the above lemma, for a Cerami sequence at level $ c_{mp} $, we get $ 2Nc_{mp}+o(1) \geq ||u_{n}||^N $. Then, we prove the next lemma similarly as \cite[Lemma 3.7]{[boer2]}, which give us a necessary way to control the exponential term for Cerami sequences in levels $ d\in (-\infty , c_{mp}] $.
\begin{lemma}\label{l10}
Let $ (u_{n})\subset X $ be a Cerami sequence for $ I $ at level $ c_{mp} $ and $ q>2N $. Then, for some $ \rho_0 >0 $ sufficiently small,
\begin{equation}\label{eq11}
\limsup\limits_{n} ||u_{n}|| < \rho_0 .
\end{equation}
\end{lemma}
\begin{proof}
By similar arguments as in \cite[Lemma 26]{[boer2]}, we obtain
$$
\limsup\limits_{n} ||u_{n}||^N  \leq \dfrac{2(q-N)}{q}\dfrac{S_{q}^{\frac{qN}{q-N}}}{(qC_q)^{\frac{N}{q-N}}} ,
$$
from which the result follows for $ C_q > 0 $ sufficiently large.
\end{proof}

\section{Proof of Theorems \ref{t1} and \ref{t2}}

We are now ready to prove our main result. However, we still need a key proposition that provide us with nontrivial critical points of $ I $ in $ X $.

\begin{proposition}\label{p1}
Let $ q > 2N $ and $ (u_{n})\subset X $ a Cerami sequence for $ I $ in level $ c_{mp} $. Then, passing to a subsequence, if necessary, only one between the alternatives occurs:

\noindent \textbf{(I)} $ ||u_{n}||\rightarrow 0 $ and $ I(u_{n})\rightarrow 0 $.

\noindent \textbf{(II)} There exists points $ y_{n} \in \mathbb{Z}^N $ such that $ y_{n} \ast u_{n} \rightarrow u $ in $ X $ for a non-trivial critical point $ u\in X $ of $ I $.
\end{proposition}
\begin{proof}
From Lemmas \ref{l1}, \ref{l33}, \ref{l8}, \ref{l10}, Propositions \ref{p3} and \ref{p2}, Corollary \ref{l11}, HLS and Hölder Inequalities and the $ (S^{+}) $ property of $ N $-Laplacian, one can prove this propositon following the same steps as in the proof of \cite[Proposition 4.1]{[boer2]}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{t1}]
\textbf{(i)} From Lemma \ref{l6} and Proposition \ref{p1} there exists a non-trivial critical point of $ I $, $ u_0 \in X $, such that $ I(u_0)=c_{mp} $.

\textbf{(ii)} Define $ K=\{v \in X\setminus \{0\} \ ; \ I'(v)=0\} $. Since $ u_0\in K $, $ K\neq \emptyset $. Thus, we can consider $ (u_{n})\subset K $ satisfying $ I(u_{n})\rightarrow c_g = \inf\limits_{v\in K} I'(v) $.

Observe that $ c_g \in [-\infty , c_{mp}] $. If $ c_g = c_{mp} $ it is done. If $ c_g < c_{mp} $, then from definition of $ K $ and Proposition \ref{p1} there exists $ (y_{n})\subset \mathbb{Z}^N $ such that $ \tilde{u}_n \rightarrow u $ in $ X $, for a non-trivial critical point $ u $ of $ I $ in $ X $ and we conclude that $ u\in K $ and
$
I(u)= c_g .
$
Particularly, we see that $ c_g > -\infty $.
\end{proof}

\begin{proof}[Proof of Corollary \ref{c400}]
Let $ (\tilde{u}_n) \subset X $ be the sequence obtained in Theorem \ref{t1} such that $ \tilde{u}_n \rightarrow u $, for $ u \in X\setminus\{0\} $,
$$
I(\tilde{u}_n)=I(u_{n}) \rightarrow c_g \mbox{ \ \ \ and \ \ \ } ||I'(u_{n})||_{X'}(1+||u_{n}||_X)\rightarrow 0 \mbox{ , as \ } n \rightarrow + \infty ,
$$
$ I(u)=c_g $ and $ I'(u)\equiv 0 $. Since $ ||\cdot|| $ is invariant under $ \mathbb{Z}^N$-translation and $ \tilde{u}_n \rightharpoonup u $ in $  W^{1,N}(\mathbb{R}^{N})  $, from the fact that any norm is weakly lower semicontinuous and by a similar argument as in Lemma \ref{l8}, we get that
$$
2Nc_g + o(1) \geq ||\tilde{u}_n||^{N} \ \ \ \Rightarrow \ \ \ 2Nc_g \geq \liminf  ||\tilde{u}_n||^{N} \geq ||u||^N > 0 .
$$
\end{proof}

To prove multiplicity, we use the genus theory, whose definition and basic properties can be found in \cite[Chapter II.5]{struwe}. As we told, the auxiliary function $ \varphi_u : \mathbb{R} \rightarrow \mathbb{R} $, given by $ \varphi_u(t) = I(tu) $, for all $ u\in X\setminus\{0\} $ and $ t \in \mathbb{R} $, enjoy the same geometry verified in \cite[Lemma 5.1]{[boer2]}, that is,

\b
(\cite[Lemma 5.1]{[boer2]}) \textbf{(a)} Let $ u\in X\setminus \{0\} $. Then, $ \varphi_u $ is even and there exists a unique $ t_u \in (0, +\infty) $ such that $ \varphi_u '(t)>0 $, for all $ t\in (0, t_u) $, and $ \varphi_u '(t)<0 $, for all $ t\in (t_u , \infty) $. Moreover, $ \varphi_u (t) \rightarrow -\infty $, as $ t \rightarrow +\infty $.

\noindent \textbf{(b)} Let $ u\in X\setminus \{0\} $. Then, there exists a unique $ t_{u}'\in (0, + \infty )$ such that $ \varphi_u(t)>0 $, for $ t\in (0, t_{u}') $, and $ \varphi_u(t)<0 $, for $ t\in (t_{u}', + \infty) $. Moreover, $ t_u $ given by item (a) is a global maximum for $ \varphi_u $.

\noindent \textbf{(c)} For each $ u\in X\setminus \{0\} $, the map $ u \mapsto t_u ' $ is continuous.
\end{lemma}

As we mentioned in the introduction, we will not explicit here all the results necessaries to prove Theorem \ref{t2}, since there are many and their proof are very similar to those one in \cite[Lemma 29 - Proposition 47]{[boer2]}. So, we only define the necessary elements that appear explicit in the proof of Theorem \ref{t2}.

Consider the Nehari's manifold for $ I $, defined by
\begin{equation}\label{VN}
\mathcal{N} = \{ u\in X \setminus \{0\} \ ; \ I'(u)(u) = 0 \},
\end{equation}
the sets
$$
I^{c}=\{u\in X \ ; \ I(u)\leq c \} \mbox{ \ , for \ } c\in \mathbb{R} \mbox{ \ and \ } D=I^0 ,
$$
and the values
$$
c_k = \inf\{c \geq 0 \ ; \ \gamma_D(I^c)\geq k \} \mbox{ \ , \ } \forall \ n\in \mathbb{N} .
$$
We can see that, in the case of $ k=1 $, we have
$ c_1 = \inf\limits_{N} I = \inf\limits_{u\in X \setminus \{0\}}\sup\limits_{t > 0}I(tu) >0 $. Moreover, we have the following two propositions that are the base for the proof of Theorem \ref{t2}.

\begin{proposition}\label{p112}
Let $ k\in \mathbb{N} $. Then, $ c_k $ is a critical value of $ I $.
\end{proposition}

\begin{proposition}\label{p113}
We have $ c_k \rightarrow + \infty $, as $ k \rightarrow + \infty $.
\end{proposition}

\begin{proof}[Proof of Theorem \ref{t2}]
By Proposition \ref{p113}, passing to a subsequence if necessary, $ c_k \rightarrow + \infty $ monotonously increasing. Thus, from Proposition \ref{p112}, there exists $ u_{k} \in X $ satisfying $ I(u_{k})=c_k $ and $ I'(u_{k})=0 $, for all $ k\in \mathbb{N} $. Since $ (c_k) $ is monotone, $ c_k >0 $, for all $ k\geq 2 $. Then, we conclude that the functions $ u_k $ are distinct and that $ u_{k} \neq 0 $, for all $ n\in \mathbb{N} $. Moreover, as $ I $ is odd, the same holds for $ -u_{k} $ and $ I(\pm u_{k}) \rightarrow + \infty $.
\end{proof}

\section{Final Considerations}

In this final section, we only want to call attention the reader that all the above arguments, included those in \cite{[boer2]} can be adapted, with minor changes, for a non-constant potential $ a: \mathbb{R}^{N} \rightarrow \mathbb{R} $ satisfying the following conditions
$$a: \mathbb{R}^{N} \rightarrow \mathbb{R} \mbox{ is continuous, } \mathbb{Z}^N\mbox{-periodic , } a\in L^{\infty}(\mathbb{R}^{N}) \mbox{ and } \inf\limits_{x\in \mathbb{R}^{N}} a(x) = a_0 > 0 . \leqno{(a_0)}$$
One also could investigate the case where the potential $ a $ is not invariant under $ \mathbb{Z}^N $ translations but is asymptotically $ \mathbb{Z}^N $-periodic, that is, there exists a $ \mathbb{Z}^N $-periodic potential $ a_p : \mathbb{R}^{N} \rightarrow \mathbb{R} $ such that $ a_p $ satisfies $ (a_0) $,
$$
0 < \inf\limits_{x\in \mathbb{R}^{N}} a(x) \leq a(x) \leq a_p(x) \ , \ \forall \ x \in \mathbb{R}^{N} \leqno{(a_1)}
$$
and
$$
\lim\limits_{|x|\rightarrow + \infty}|a(x) - a_p(x)| = 0 . \leqno{(a_2)}.
$$
To do that, one can argue as in \cite{[alves2]}. \\

\noindent \textbf{Acknowledgements:} The first author was supported   by  Coordination of Superior Level Staff Improvement-(CAPES)-Finance Code 001 and  S\~ao Paulo Research Foundation-(FAPESP), grant $\sharp $ 2019/22531-4, while the second  author was supported by  National Council for Scientific and Technological Development -(CNPq),   grant $\sharp $ 307061/2018-3 and FAPESP  grant $\sharp $ 2019/24901-3.

redpajama_set_name: RedPajamaArXiv


During this time of year we like to use the word "tradition" a lot - passing down activities that older generations took part in or creating new ones with our own families. Using specific things like grandmas fine china or hanging an heirloom ornament on the tree (if you have one) gives us the warm and fuzzies. I was asked to share my opinion on holiday traditions for baby's first Christmas recently and during the conversation I found myself talking about the importance of creating something that is comfortable for your individual family - however that looks. Gone are the days when everyone lived in the same state let alone neighborhood, so getting together is harder and it can make the holidays a little tricky. My family always had the same tradition during the holidays, Christmas Eve was at my parent's house with family and friends and Christmas Day was at our friend's house. We did that for many years and then the kids started getting married and having kids of their own and the holiday's started to look different. After I moved back to New York, we made the decision to continue to go to Florida for the holidays. Once we had a family of our own we decided to continue that tradition with our children with a twist. Now we split it! We still go to Florida to see our friends and family, and we also have our own family Christmas at home. This way we get to create new family traditions and continue with the old ones, (and we don't have to drag gifts with us).
I got to thinking about how holidays look now that we're parents ourselves. My parents, mom especially, wanted the holidays to be full of love, food and gift giving. We used to decorate the tree as a family when we were kids and my mom made it a point to make sure my brother and I had the same amount of boxes under the tree as to not disappoint anyone and she still does it with her grandchildren! (crazy much?) I like to do a lot of similar things that I was brought up with. We decorate our tree and home, bake cookies to give to friends and donate to charities. I feel like now a days the holidays are much faster moving. Maybe it's because there's a finite amount of time to spend with our loved ones because we don't live near one another that we're trying to cram it all in? I dunno, it just feels like it happens way too quickly. Don't blink, you might miss it!
As time passes, some traditions may fall by the wayside. I was chatting with a friend the other day who is not particularly fond of putting up a Christmas tree. It's not a big deal to her and she figures her daughter is still small enough that she won't know any better. She does however, have an artificial tree that was given to her by a neighbor that her nanny found last week and took it upon herself to put up and decorate while my friend was at work. She said when she came home she was taken off guard and pleasantly surprised by the gesture. It's not that she doesn't like having a tree she realized, she doesn't like assembling it! So we talked about creating a tradition by where her daughter gets to put up the tree with her nanny and that can be their special time. I likened it to creating a recipe - put in all of the ingredients that make you feel good and remove the ones that don't. Every family recipe will look different - unique to them, and it can evolve. Some recipes over time increase the ingredients, some switch out for new ones. There's no wrong way to do it.
My family recipe is still in the works. We're constantly tweaking it. Some years we try a new thing, and other years we stick to the familiar plan. That's the great thing about recipes, you can change them. I can't help but wonder how our recipe will evolve as our children grow. Sometimes the thought of losing the familiarity of a tradition is a little sad, other times a change can be exciting. I'm choosing to embrace all of our traditions and creating new ones with my family.
What does your family recipe look like?


Howdy! Allowed in order to my weblog, with this time I'll teach you in relation to American Blind And Wallpaper.
So, if you desire to receive all of these fantastic pics related to American Blind And Wallpaper, click save icon to download the pics to your computer.
These are available for download, if you want and want to get it, click save badge on the article, and it'll be directly saved in your laptop computer.
As a final point if you need to get unique and latest photo related to American Blind And Wallpaper, please follow us on google plus or bookmark this website, we try our best to offer you daily up-date with all new and fresh graphics. We do hope you like keeping right here. For most upgrades and recent information about American Blind And Wallpaper images, please kindly follow us too, or you can book mark this page on bookmark area. We attempt to offer you update periodically with fresh and new shots, love your browsing, and find the ideal for you.
Hey, take your seat and lets watch this video. Enjoy the show!


And learn to sing a song.
They're just as fun for boys.
He's such a cute little guy.
All I want to know is happiness.
I yearn to be five again some way, to know innocence, pure and hope, uncrushed, as the sticky bouquet of a dear boy's first love.
I desire to relive the moment before a forced letter was written, time was lost and my fate was sealed by stalking Evil.
I crave to lick ice cream off a stick and not worry about dripping it on the couch.
I want to play in a sloppy puddle of mud, throw myself into a crunchy pile of leaves as they fall, draw funny chalk figures on a sidewalk and believe again that I can change the future.
I hunger to feast on fresh baked brownies and solid chocolate Easter bunnies because they melt brown sweet goodness in my mouth and fill both my eyes and tummy.
I long to play in the sun without thinking of wrinkles, fill scrapbooks and boxes of memories with the reds, blues, greens and purples of pipe cleaner reindeer and then tie them up with ribbons that twizzle and sparkle with silvery glitter.
I wish to return to a time when life and joy were simple, doors and windows could be left unlatched, neighbors were safe havens, strangers were not a danger, the use of drugs consisted of two aspirin and, "Call me in the morning," trust between friends was never violated and play was freedom during a light spring drizzle.
I ache to hope that anything is possible, that nightmares aren't real and that birthday wishes come true.
I strive to be oblivious to the atrocities of life, thrilled again over the beauty of the earth, the crisp scent of a season's shower or the powdery formations of the mists in a cloud.
I request the world not consist of deception, depressing news, illness or broken, abandoned hearts, the inconceivable theft of one child or the violent murder of another.
I pray one day I'll kiss, caress, tuck my children's' children into their beds and teach them their prayers to my God not forsaken.
I choose to believe in the power of forgiveness, truth, justice, peace, reconciliation, man, child and woman kind, and listening to my little boy angel as he whispers ever near.
I offer my soul, my heart, my brain, bared for all to judge, to see and to read, my bones gnarled from the battles of dreams.
It's time again for another birth.
Is HEAVEN'S VERY SPECIAL CHILD.


For the Minimalism Game Day 25, I was determined to get everything from the kitchen. I want to reorganize my kitchen shelves, and I realize I have too much stuff, mostly stuff that I never use. A lot of these items came from a top shelf that I can't reach and I usually just put stuff there when I want to forget it.
Not sure this numbering will be in any understandable order, but I will try to go from left to right.
An expired hand sanitizer (I never use hand sanitizer).
A woven item made to hold paper plates that my parents used in the 1980s. I don't use paper plates and this had gross food stuck in it (probably from the 1980s).
A metal thing that been sitting on my sink for six years (I guess I don't need it).
A candle that I moved with from my old apartment six years ago and still haven't lit.
(back row) Expired jelly from fridge.
Set of creamer and sugar holders on a tray that was my mother's and I hate and I hate looking at and why have I looked at it for the last 17 years?
Second piece of that set.
Third piece of that set.
I am not sure the intended purpose of the green thing, where it came from, or how long I've had it.
Ice cream holder–I don't need a dedicated glass for this purpose and haven't used in at least six years.
Ball jar with no lid.


text:
Astragalus pseudohypogaeus är en ärtväxtart som beskrevs av S.B.Ho. Astragalus pseudohypogaeus ingår i släktet vedlar, och familjen ärtväxter. Inga underarter finns listade i Catalogue of Life.

Källor

Vedlar
pseudohypogaeus

redpajama_set_name: RedPajamaWikipedia