text:
\section{Introduction and main result}
\noindent
Let $N>p>1$.\ In two seminal papers, T.\,Aubin \cite{Au} and G.\,Talenti \cite{talenti} showed that the minimizers of the Sobolev quotient
\begin{equation}
\label{opt-local}
\mathcal{S}_{p}=\inf_{u\in D^{1,p}(\mathbb{R}^N)\setminus\{0\}} \frac{\|\nabla u\|^p_{L^p(\mathbb{R}^N)}}{\displaystyle\left(\int_{\mathbb{R}^N} |u|^\frac{N\,p}{N-p}\, dx\right)^\frac{N-p}{N}},
\end{equation}
are given by the family of functions
\begin{equation}
\label{aubintalenti}
U_{t}(x)={\mathcal C}\,t^\frac{p-N}{p}\,U\left(\frac{x-x_0}{t}\right),
\qquad {\mathcal C}\in\mathbb{R}\setminus\{0\},\ t>0,\ x_0\in\mathbb{R}^N,
\end{equation}
where
\[
U(x)=\mathcal{C}_{N,p}\,\big(1+|x|^{\frac{p}{p-1}}\big)^{\frac{p-N}{p}},\qquad\qquad \mathcal{C}_{N,p}=\left(\int_{\mathbb{R}^N}\big(1+|x|^{\frac{p}{p-1}}\big)^{-N}\,dx\right)^\frac{N\,p}{p-N}.
\]
For the limit case $p=1$, the problem was investigated by H.\,Federer and W.\,H.\,Fleming
in \cite{federer} and by V.\,G.\,Maz'ya in \cite{mazia}.
\par
On one side, these results establish an enlightening  connection between the theory of Sobolev spaces and the theory of classical isoperimetric inequalities.
On the other side, they provide a very powerful tool for the study of second order partial differential equations involving nonlinearities reaching
the critical growth with respect to the Sobolev embedding.
In the case $p=2$, these classification results were formally derived by G.\,Rosen in \cite{Ro}.
\par
The variational problem \eqref{opt-local} is related to the following equation involving the $p-$Laplace operator $\Delta_p u={\rm div}(|\nabla u|^{p-2}\nabla u)$,
\begin{equation}
\label{pEnt-loc}
-\Delta_p u=|u|^{\frac{N\,p}{N-p}-2}\,u, \qquad \mbox{ in }\mathbb{R}^N.
\end{equation}
In fact, a nontrivial problem is that of proving that the only {\em fixed sign} solutions of this equation are precisely given by \eqref{aubintalenti}, for a suitable choice of the constant.
\par
In the restricted class of {\em radially symmetric fixed sign} solutions to \eqref{pEnt-loc},
this was shown by M.\,Guedda and L.\,Veron in \cite{GueddaV}. Recently, in \cite[Corollary 1.3]{Vetois} for the case $1<p\leq 2\,N/(N+2)$,
in \cite[Theorem 1.2]{ClassDino} for the case $2\,N/(N+2)<p\leq 2$ and in \cite[Theorem 1.1]{sciunziCl} for the case $2<p<N$, it was proved that any positive weak solution to \eqref{pEnt-loc} is radially symmetric and radially decreasing about some point,
thus answering positively to the classification of constant sign solutions to \eqref{pEnt-loc}.
\par
The result by Aubin and Talenti, as well as the previous results in the linear case $p=2$, strongly rely on the reduction of the problem
 to an {\em ordinary differential equation} which can be explicitly solved. We recall that more recently, the Aubin-Talenti result has been reproved in \cite[Theorem 2]{CENV} by means of very different techniques, based on Optimal Transport.
\vskip.2cm
\noindent
Let now $s\in (0,1)$, $p>1$ and $N>s\,p$. The goal of this paper is to provide information about the asymptotic behavior at infinity of
optimizers of the problem
\begin{equation}
\label{costanteintro}
\mathcal{S}_{p,s}:=\inf_{u\in D^{s,p}(\mathbb{R}^N)\setminus\{0\}} \frac{\displaystyle\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dx\,dy}{\displaystyle\left(\int_{\mathbb{R}^N} |u|^\frac{N\,p}{N-s\,p}\,dx\right)^\frac{N-s\,p}{N}},
\end{equation}
which is related to the {\it fractional} Sobolev embedding, see for example \cite[Theorem 1]{MS}. Here
\[
D^{s,p}(\mathbb{R}^N)=\left\{u\in L^{{N\,p/(N-s\,p)}}(\mathbb{R}^N):
\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dx\,dy<\infty\right\}.
\]
In the limit case $p=1$, the sharp constant above has been determined in \cite[Theorem 4.1]{FS}
(see also \cite[Theorem 4.10]{BraLinPar}). The relevant extremals are given by characteristic functions of balls, exactly as in the local case.
\par
Problem \eqref{costanteintro} for $p>1$ is now related to the study of the nonlocal integro-differential equation
\begin{equation}
\label{quazione}
(-\Delta_p)^s u=|u|^{\frac{N\,p}{N-s\,p}-2}\,u,\qquad \mbox{ in }\mathbb{R}^N,
\end{equation}
where, formally, the operator $(-\Delta_p)^s$ is defined on smooth functions as
\[
(-\Delta_p)^s u(x) = 2\, \lim_{\varepsilon \searrow 0} \int_{\mathbb{R}^N \setminus B_\varepsilon(x)} \frac{|u(x) - u(y)|^{p-2}\, (u(x) - u(y))}{|x - y|^{N+s\,p}}\, dy, \qquad x \in \mathbb{R}^N.
\]
This operator appears in some recent works like \cite{AMRT} and \cite{IN}. See also \cite{DKP2,ILPS,IMS,IMS2,KMS1} and the references therein for some existence and regularity results.
\par
In the Hilbertian case $p=2$, it is known by \cite[Theorem 1.1]{Cotsiolis} that the family of functions
\begin{equation}
\label{cotsiolis}
U_{t}(x)={\mathcal C}\,t^\frac{2\,s-N}{2}\,\left(1+\left(\frac{|x-x_0|}{t}\right)^2\right)^{\frac{2\,s-N}{2}},\qquad \mathcal{C}\in\mathbb{R}\setminus\{0\},\ t>0,\ x_0\in\mathbb{R}^N,
\end{equation}
is the only set of minimizers for the best Sobolev constant $\mathcal{S}_{2,s}$. More precisely, in \cite[Theorem 1.1]{Cotsiolis} it is proved that the family \eqref{cotsiolis} provides all the minimizers of the following problem
\[
\widetilde{\mathcal{S}}_{2,s}:=\inf_{u\in D^{s,2}(\mathbb{R}^N)\setminus\{0\}} \frac{\big\|(-\Delta)^{s/2}\,u\big\|_{L^2(\mathbb{R}^N)}^2}{\displaystyle\left(\int_{\mathbb{R}^N} |u|^\frac{2\,N}{N-2\,s}\,dx\right)^\frac{N-2\,s}{N}},
\]
where the $L^2$ norm of $(-\Delta)^{s/2} u$ is defined in terms of the Fourier transform. By using \cite[Proposition 3.6]{Guida}, one knows that
\[
\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2\,s}}\,dx\,dy=c\, \big\|(-\Delta)^{s/2}\,u\big\|_{L^2(\mathbb{R}^N)}^2,
\]
for some $c=c(N,s)>0$. This implies that \eqref{cotsiolis} are the only solutions of \eqref{costanteintro} as well.
\par
It is also known by \cite[Theorem 1]{CLO} that, for a suitable
positive constant ${\mathcal C}={\mathcal C}(N,s)$, \eqref{cotsiolis} are the only positive solutions of
\begin{equation}
\label{linear-crit}
(-\Delta)^s u=|u|^{\frac{2\,N}{N-2\,s}-2}\,u \qquad \mbox{ in }\mathbb{R}^N.
\end{equation}
The result in \cite{CLO} is based upon the full equivalence between the weak solutions to \eqref{linear-crit} and the integral formulation
\begin{equation}
\label{integral-f}
u(x)=\int_{\mathbb{R}^N}\frac{|u(y)|^{\frac{2\,N}{N-2\,s}-2}\,u}{|x-y|^{N-2\,s}}\,dy,\qquad u\in L^{\frac{2\,N}{N-2\,s}}(\mathbb{R}^N),
\end{equation}
on the validity of some Kelvin transform and on moving plane arguments applied to \eqref{integral-f}, in the spirit of \cite{lieb}.
\vskip.2cm
Unfortunately, in the nonlocal and nonlinear case $p\neq 2$ there is {\em no} Kelvin transform and {\em no}
equivalent integral representation result.
Furthermore, even restricting to the class of radially symmetric functions, establishing a classification result for the optimizers
of \eqref{costanteintro} seems very hard.
We conjecture that
the optimizers are given by
\begin{equation}
\label{pTal}
U_{t}(x)={\mathcal C}\,t^\frac{s\,p-N}{p}\,U\left(\frac{x-x_0}{t}\right),
\qquad {\mathcal C}\in\mathbb{R}\setminus\{0\},\ t>0,\ x_0\in\mathbb{R}^N,
\end{equation}
where this time
\begin{equation}
\label{famloc}
U(x):=\mathcal{C}_{N,p,s}\,\big(1+|x|^{\frac{p}{p-1}}\big)^{\frac{s\,p-N}{p}},\qquad\qquad \mathcal{C}_{N,p,s}=\left(\int_{\mathbb{R}^N}\big(1+|x|^{\frac{p}{p-1}}\big)^{-N}\,dx\right)^\frac{N\,p}{s\,p-N}.
\end{equation}
Notice that \eqref{pTal} and \eqref{famloc} are consistent with the cases $p=2$ or $s=1$, in the last case we are back to the family of Aubin-Talenti functions \eqref{aubintalenti}
for the $p-$Laplacian operator.
\par
In the main result of this paper, we prove that extremals for \eqref{costanteintro} have exactly the decay rate at infinity dictated by formula \eqref{famloc}. Namely, we have the following.
\begin{theorem}
\label{sharp-Dec}
Let $U\in D^{s,p}(\mathbb{R}^N)$ be any minimizer for \eqref{costanteintro}.
Then $U\in L^\infty(\mathbb{R}^N)$ is a constant sign, radially symmetric and monotone function with
\begin{equation}
\label{sciarpa2}
\lim_{|x|\to\infty}|x|^{\frac{N-s\,p}{p-1}}U(x)=U_\infty,
\end{equation}
for some constant $U_\infty\in\mathbb{R}\setminus\{0\}$.
\end{theorem}
\begin{remark}
\label{referee}
As it will be apparent from the proof of Theorem \ref{sharp-Dec}, the same conclusion \eqref{sciarpa2} can be drawn for any constant sign, radially symmetric and monotone solution of the critical equation \eqref{quazione}. In the local case this property is a plain consequence of the aforementioned classification result of \cite{GueddaV}.
\end{remark}
The building blocks of Theorem~\ref{sharp-Dec} are a weak $L^q$ estimate for the minimizers (Proposition \ref{lp-weak}), a Radial Lemma for Lorentz spaces (Lemma \ref{lm:border}) and the fact that the function
$$
\Gamma(x):=|x|^{-\frac{N-s\,p}{p-1}},\quad x\in\mathbb{R}^N\setminus\{0\},
$$
is a weak solution of $(-\Delta_p)^su=0$ in $\mathbb{R}^N\setminus B_r$, for any $r>0$ (Theorem \ref{fundamental}). Then the crucial point will be constructing
suitable barrier functions to be combined with a version of the comparison principle for $(-\Delta_p)^s$
recently obtained in \cite{IMS}.
Observe that for $s=1$, the function $\Gamma$ above is nothing but the fundamental solution of the $p-$Laplacian.
\par
We wish to stress that Theorem~\ref{sharp-Dec} also provides  a very
useful tool for the investigation of {\em existence} of weak solutions for the nonlocal Brezis-Nirenberg problem in
a smooth bounded domain $\Omega\subset\mathbb{R}^N$, i.e.
\[
\begin{cases}
(-\Delta_p)^s\, u  =\lambda\,|u|^{p-2}\,u+|u|^{\frac{N\,p}{N-s\,p}-2}u & \mbox{ in } \Omega, \\
u  = 0 & \mbox{ in }\mathbb{R}^N \setminus \Omega,
\end{cases}
\]
where $\lambda> 0$. This problem has been studied in \cite{SV} for $p=2$. For a general exponent $1<p<N/s$, by means of \eqref{sciarpa2}, one can estimate truncations of $U_t$
via a suitable cut-off function in terms of the sharp
constant $\mathcal{S}_{p,s}$ {\em without knowing the explicit form} of the optimizers. Such a procedure is new even for the local case.
These estimates allow to apply mountain pass or linking arguments by forcing
the min-max levels to fall inside a compactness range for the energy functionals, see \cite{per-mosc-squ-yang} for more details.

\begin{plan}
In Section \ref{sec:2} we set all the notations, definitions and basic facts that will be needed throughout the paper. Then in Section \ref{sec:3} we prove existence of solutions for \eqref{costanteintro}, together with some basic integrability properties. We also prove that extremals have to be comparable to
$$
x\mapsto |x|^{-\frac{N-s\,p}{p-1}},\quad x\in\mathbb{R}^N\setminus\{0\},
$$
at infinity (Corollary \ref{lower-Dec}). Then the exact behavior \eqref{sciarpa2} is proved in Section \ref{sec:4}. The paper ends with Appendix \ref{sec:appA}, containing a rigourous computation of the fractional $p-$Laplacian of a power function.
\end{plan}

\begin{acknow}
We warmly thank Yannick Sire for some informal discussions on the subject of this paper. We owe Remark \ref{referee} to the kind courtesy of an anonymous referee, we wish to thank him.
This research has been partially supported by {\it Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni} (INdAM) and by {\it Agence Nationale de la Recherche}, through the project ANR-12-BS01-0014-01 {\sc Geometrya}.
Part of this paper was written during a visit of S.\,M. and M.\,S. in Marseille in March 2015. The I2M and FRUMAM institutions are gratefully acknowledged.
\end{acknow}

\section{Preliminary results}

\label{sec:2}

\subsection{Notation}
In the following we will fix $s\in (0, 1)$, $p>1$ and $N$ as the dimension, letting for brevity
\[
p^*=\frac{N\,p}{N-s\,p}.
\]
We denote by $\omega_N$ the measure of the $N-$dimensional ball having unit radius.
Moreover, $\mathbf{S}^{N-1}$ will denote $\{x\in \mathbb{R}^N:|x|=1\}$. For $E\subseteq \mathbb{R}^N$ measurable we denote by $|E|$ its $N-$dimensional Lebesgue measure, by $E^c=\mathbb{R}^N\setminus E$ its complement and by $\chi_E$ its characteristic function. If $u:E\to \mathbb{R}$ is measurable we set
\[
[u]_{W^{s, p}(E)}^p:=\int_{E\times E}\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\, dx\, dy, \qquad [u]_{s,p}:=[u]_{W^{s,p}(\mathbb{R}^N)},
\]
and for any $q>0$
\[
\|u\|_{L^q(E)}:=\left(\int_{E}|u|^q\, dx\right)^{1/q},\qquad\|u\|_q:=\|u\|_{L^q(\mathbb{R}^N)}.
\]
Finally, for $t\in \mathbb{R}$ we will use the notation
\[
J_p(t)=|t|^{p-2}\,t.
\]

\subsection{Elementary inequalities}

We list here some useful inequalities on the function $J_p$. First,
consider the case $p\geq 2$. We recall that
\begin{equation}
\label{licciz}
|J_p(a)-J_p(b)|\le (p-1)\,(|a|^{p-2}+|b|^{p-2})\,|a-b|,\qquad a,b\in\mathbb{R},\quad p\ge 2,
\end{equation}
as a consequence of the mean value Theorem.
In \cite[eq. (2.7)]{IMS} it is also proved the following inequality
\begin{equation}
\label{27}
J_p(a)-J_p(a+b)\leq -2^{2-p}\,b^{p-1},\qquad a\in \mathbb{R}, \ b\geq 0,\quad p\geq 2.
\end{equation}
Let us consider the case $p\in (1,2]$.
We recall the well-known monotonicity inequality
\begin{equation}
\label{seconda}
\big(J_p(a)-J_p(b)\big)\,(a-b)\ge c\,\frac{|a-b|^2}{(a^2+b^2)^\frac{2-p}{2}},\qquad a,b\in\mathbb{R}\setminus\{0\},\quad p\in(1,2].
\end{equation}
Next we prove the following inequality
\begin{equation}
\label{ineqb}
J_p(a)-J_p(a-b)\geq \max\left\{J_p(A)-J_p(A-b),\, \left(\frac{b}{2}\right)^{p-1}\right\},\quad a\in [0,A], b\geq 0,\ p\in (1,2].
\end{equation}
We distinguish two cases. First suppose that $a\geq b/2$. The function $t\mapsto J_p(t)-J_p(t-b)$ is readily seen to be decreasing on $[b/2, +\infty[$, so that
\[
J_p(a)-J_p(a-b)\geq J_p(A)-J_p(A-b)
\]
in this case.
On the other hand, if $a<b/2$, being $J_p$ odd and increasing we have
\[
J_p(a)-J_p(a-b)\geq J_p(b-a)\geq J_p\left(\frac{b}{2}\right),
\]
and thus \eqref{ineqb} is proved.

\subsection{Functional framework}

We consider the space
\[
D^{s,p}_0(\Omega):=\Big\{u\in L^{p^*}(\Omega): \text{$u\equiv 0$ in $\Omega^c$},\,\, [u]_{s,p}<+\infty\Big\},\qquad
D^{s,p}(\mathbb{R}^N):=D^{s,p}_0(\mathbb{R}^N),
\]
which is a Banach space with respect to the norm $[\,\cdot\, ]_{s,p}$.  Our first aim is to prove, under suitable regularity assumptions on $\partial\Omega$, that $C^\infty_c(\Omega)$ is dense in  $D^{s,p}_0(\Omega)$
with respect to the norm $[\,\cdot \, ]_{s,p}$.
While this density result is well-known for $D^{s,p}_0(\Omega)\cap L^p(\Omega)$ (see for example \cite{FSV}), we will need to remove the $L^p$ assumption in the following. Finally we will prove a comparison principle in a rather general space.

\begin{theorem}
\label{density}
Let $\Omega\subset\mathbb{R}^N$ be an open set such that $\partial\Omega$ is compact and locally the graph of a continuous function.  Then $D^{s,p}_0(\Omega)$ is the completion of $C^\infty_c(\Omega)$ with respect to the norm $[\,\cdot\,]_{s,p}$.
\end{theorem}

\begin{proof}
Let $u\in D^{s,p}_0(\Omega)$. Reasoning on  $u_+$ and $u_-$ separately (which still belong to $D^{s,p}_0(\Omega)$), we can suppose that $u$ is nonnegative. Consider, for $\varepsilon>0$, the function $u_\varepsilon=(u-\varepsilon)_+$. Using the $1-$Lipschitzianity of $t\mapsto (t-\varepsilon)_+$ it is readily checked that
\[
|u_\varepsilon(x)-u_\varepsilon(y)|^p \leq |u(x)-u(y)|^p, \qquad |u_\varepsilon(x)-u_\varepsilon(y)|^p \to |u(x)-u(y)|^p,\ \text{ a.e. in $\mathbb{R}^{2N}$}.
\]
Therefore $u_\varepsilon\in D^{s,p}_0(\Omega)$ and by dominated convergence $[u_\varepsilon]_{s,p}\to [u]_{s,p}$. This in turn implies that $u_\varepsilon\to u$ in $D^{s,p}_0(\Omega)$ by uniform convexity of the norm. Now Chebyshev's inequality  ensures that $\mathrm{supp}(u_\varepsilon)$ has finite measure, thus by H\"older's inequality we get $u_\varepsilon\in L^p(\mathbb{R}^N)$. This yields
\[
u_\varepsilon\in D^{s,p}_0(\Omega)\cap L^p(\mathbb{R}^N),
\]
and \cite[Theorem 6]{FSV} ensures that $u_\varepsilon$ can be approximated,
in the norm $[\, \cdot\, ]_{s,p},$ by functions which belong to $C^\infty_c(\Omega)$.
\end{proof}

\noindent
We recall the following nonlocal Hardy inequality proved in  \cite[Theorem 2]{FS}.

\begin{proposition}[Hardy's inequality]
Let $N>s\,p$. Then there exists $C=C(N, p, s)>0$ such that
\begin{equation}
\label{hardyin}
\int_{\mathbb{R}^N}\frac{|u|^p}{|x|^{s\,p}}\, dx\leq C\,[u]_{s, p}^p,\qquad \mbox{ for every } u\in D^{s,p}(\mathbb{R}^N).
\end{equation}
\end{proposition}

\noindent
We then define a suitable space where a comparison principle holds true. For any $\Omega\subset \mathbb{R}^N$ open set, we define
\[
\begin{split}
\widetilde{D}^{s,p}(\Omega):=\Big\{u\in L^{p-1}_{\mathrm{\mathrm{loc}}}(\mathbb{R}^N)\cap L^{p^*}(\Omega) \,:\, &\exists\, E\supset\Omega \text{ with $E^c$ compact},\, \mathrm{dist}(E^c, \Omega)>0  \\
&\mbox{ and }\,[u]_{W^{s,p}(E)}<+\infty \Big\}.
\end{split}
\]
We wish to point out that the definition above is given having in mind the case of $\Omega$ being an exterior domain, i.e. the complement of a compact set.
Essentially, we consider functions $u$ which are regular in a slight enlargement
of $\Omega$ and possibly rough far from $\Omega$.
\vskip.2cm
The following expedient result will be used in the sequel.
\begin{lemma}[Nash-type interpolation inequality]
Let $1<p<\infty$ and $0<s<1$. For every $u\in L^{p-1}(B_R)$ such that $[u]_{W^{s,p}(B_R)}<+\infty$ we have
\begin{equation}
\label{disuguaglianza}
\|u\|^p_{L^p(B_R)}\le C\,R^{s\,p}\, [u]^p_{W^{s,p}(B_R)}+\frac{C}{R^\frac{N}{p-1}}\,\|u\|^p_{L^{p-1}(B_R)},
\end{equation}
for some $C=C(N,s,p)>0$.
\end{lemma}
\begin{proof}
We observe that it is enough to prove \eqref{disuguaglianza} for $R=1$, then the general case can be obtained with a simple scaling argument.
\par
At first, we prove \eqref{disuguaglianza} for functions in $W^{s,p}(B_1)$. We can use a standard compactness argument: assume by contradiction that \eqref{disuguaglianza} is false on $W^{s,p}(B_1)$, then there exists a sequence $\{u_n\}_{n\in\mathbb{N}}\subset W^{s,p}(B_1)$ such that
\begin{equation}
\label{azzurdo}
\|u_n\|^p_{L^p(B_1)}=1\qquad \mbox{ and }\qquad [u_n]^p_{W^{s,p}(B_R)}+\|u_n\|^p_{L^{p-1}(B_1)}\le \frac{1}{n}.
\end{equation}
In particular, the sequence is bounded in $W^{s,p}(B_1)$. Thus by compactness of the embedding $W^{s,p}(B_1)\hookrightarrow L^{p}(B_1)$ (see for example \cite[Theorem 7.1]{Guida}) we get that (up to a subsequence) it converges strongly in $L^p(B_1)$ to $u\in W^{s,p}(B_1)$. From \eqref{azzurdo} we now easily get a contradiction. This shows that \eqref{disuguaglianza} is true for functions in $W^{s,p}(B_1)$.
\par
We now take $u\in L^{p-1}(B_1)$ with finite Gagliardo seminorm. Observe that
\begin{equation}
\label{valorassoluto}
\big||u(x)|-|u(y)|\big|\le |u(x)-u(y)|,
\end{equation}
so that
\[
\big[|u|\big]_{W^{s,p}(B_1)}\le [u]_{W^{s,p}(B_1)}.
\]
Thus we can assume $u$ to be positive without loss of generality. We define the increasing sequence $u_n=\min\{u,n\}\in W^{s,p}(B_1)$. From the first part of the proof and $1-$Lipschitzianity of the function $t\mapsto \min\{t,n\}$ we have
\[
\begin{split}
\|u_n\|^p_{L^p(B_1)}&\le C\,[u_n]^p_{W^{s,p}(B_1)}+C\,\|u_n\|^p_{L^{p-1}(B_1)}\\
&\le C\,[u]^p_{W^{s,p}(B_1)}+C\,\|u_n\|^p_{L^{p-1}(B_1)}.
\end{split}
\]
Passing to the limit and using the Monotone Convergence we get the desired conclusion.
\end{proof}
\begin{lemma}
\label{lm:frustone}
Let $1<p<\infty$ and $0<s<1$. For every $u\in L^{p-1}_{\mathrm{\mathrm{loc}}}(\mathbb{R}^N)$, every $E\subset \mathbb{R}^N$ open set and every ball $B_R\subset E$, we have
\begin{equation}
\label{frustone}
\int_{E} \frac{|u(x)|^{p}}{(1+|x|)^{N+s\,p}}\, dx\le C\,[u]^p_{W^{s,p}(E)}+C\,\|u\|^p_{L^{p-1}(B_R)},
\end{equation}
for some $C=C(N,p,s,R)>0$, blowing-up as $R\searrow 0$.
\end{lemma}
\begin{proof}
We assume that the right-hand side on \eqref{frustone} is finite, otherwise there is nothing to prove. For simplicity, we can suppose that $B_R$ is centered at the origin. From \eqref{disuguaglianza}, we infer
\begin{equation}
\label{disuguaglianza2}
\int_{B_R} \frac{|u|^p}{(1+|x|)^{N+s\,p}}\,dx\le C\,R^{s\,p}\, [u]^p_{W^{s,p}(B_R)}+\frac{C}{R^\frac{N}{p-1}}\,\|u\|^p_{L^{p-1}(B_R)}.
\end{equation}
On the smaller ball $B_{R/2}$ (still centered at the origin), we have
\[
\int_{E\setminus B_R} \int_{B_{R/2}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dy\,dx\le [u]^p_{W^{s,p}(E)}<+\infty.
\]
Since
\[
|x-y|\le \frac{3}{2}\, |x|,\qquad x\in E\setminus B_R,\ y\in B_{R/2},
\]
we get
\[
\begin{split}
\int_{E\setminus B_R} \int_{B_{R/2}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dy\,dx&\ge c\,R^N\,\int_{E\setminus B_R} \frac{|u|^p}{|x|^{N+s\,p}}\,dx\\
&-c\, \left(\int_{E\setminus B_R} \frac{1}{|x|^{N+s\,p}}\,dx\right)\, \int_{B_{R/2}} |u|^p\,dy.
\end{split}
\]
In conclusion, the previous estimate proves
\[
\int_{E\setminus B_R} \frac{|u|^p}{(1+|x|)^{N+s\,p}}\,dx\le \frac{C}{R^N}\,[u]_{W^{s,p}(E)}^p+\frac{C}{R^{N+s\,p}}\, \int_{B_{R/2}} |u|^p\,dx.
\]
Using \eqref{disuguaglianza} to estimate the $L^p$ norm in the right-hand side gives
\begin{equation}
\label{disuguaglianza3}
\int_{E\setminus B_R} \frac{|u|^p}{(1+|x|)^{N+s\,p}}\,dx\le \frac{C}{R^N}\,\,[u]_{W^{s,p}(E)}^p+\frac{C}{R^{N\,\frac{p}{p-1}+s\,p}}\, \left(\int_{B_{R/2}} |u|^{p-1}\,dy\right)^\frac{p}{p-1},
\end{equation}
possibly for a different constant $C=C(N,s,p)>0$.
By summing up \eqref{disuguaglianza2} and \eqref{disuguaglianza3} we get the conclusion.
\end{proof}
\noindent
The next proposition shows that in the space $\widetilde{D}^{s, p}(\Omega)$, the operator $(-\Delta_p)^s$ is well defined.

\begin{proposition}
\label{deltaps}
For any $u\in \widetilde{D}^{s,p}(\Omega)$, the operator
\[
D^{s,p}_0(\Omega)\ni \varphi\mapsto  \langle (-\Delta_p)^s u, \varphi\rangle:=\int_{\mathbb{R}^N\times \mathbb{R}^N}\frac{J_p(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{N+s\,p}}\, dx\, dy
\]
is well defined and belongs to the dual space $(D^{s, p}_0(\Omega))^*$.
\end{proposition}

\begin{proof}
We proceed as in \cite[Lemma 2.3]{IMS}. Let $E\supset \Omega$ be such that $E^c$ is compact, ${\rm dist}(E^c, \Omega)>0$ and $[u]_{W^{s,p}(E)}<+\infty$.
Since $\varphi\equiv 0$ in $\Omega^c$, we split the integral as
\[
\begin{split}
\int_{\mathbb{R}^N\times \mathbb{R}^N}&\frac{J_p(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{N+s\,p}}\, dx\, dy\\
&=\int_{E\times E}\frac{J_p(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{N+s\,p}}+2\int_{\Omega\times E^c}\frac{J_p(u(x)-u(y))\,\varphi(x)}{|x-y|^{N+s\,p}}\, dx\, dy.
\end{split}
\]
By H\"older's inequality the first term is finite and defines a continuous linear functional on $D^{s,p}_0(\Omega)$. Let us focus on the second one. By using that $\varphi\equiv 0$ in $E^c$, we need to show that
\[
\varphi\mapsto \int_{\Omega}\varphi(x)\left(\int_{E^c} \frac{J_p(u(x)-u(y))}{|x-y|^{N+s\,p}}\, dy\right) dx,
\]
is a continuous linear functional on $D^{s,p}_0(\Omega)$. By means of Hardy's inequality \eqref{hardyin}, we get that convergence of $\{\varphi_n\}_{n\in\mathbb{N}}$ in $D^{s,p}_0(\Omega)\subset D^{s,p}(\mathbb{R}^N)$ implies strong convergence in $L^p(\Omega)$ of $\{|x|^{-s}\varphi_n\}_{n\in\mathbb{N}}$. Thus to prove the claim it suffices to show that
\[
x\mapsto |x|^s\int_{E^c} \frac{J_p(u(x)-u(y))}{|x-y|^{N+s\,p}}\, dy\in L^{p'}(\Omega).
\]
Being $E^c$ compact and ${\rm dist}(E^c, \Omega)\geq \delta>0$ it holds
\begin{equation}
\label{xmy}
|x-y|\geq C\,(1+|x|), \qquad \mbox{ for every } x\in \Omega,\ y\in E^c,
\end{equation}
for some $C=C(E,\Omega)>0$. Thus, for almost every $x\in\Omega$, we can estimate
\[
\left|\int_{E^c} \frac{|x|^s\,J_p(u(x)-u(y))}{|x-y|^{N+sp}}\, dy\right|\leq C\left[|E^c|\,\frac{|u(x)|^{p-1}}{(1+|x|)^{\frac{N+sp}{p'}}}+\frac{1}{(1+|x|)^{N+s\,(p-1)}}\,\int_{E^c}|u|^{p-1}\, dy\right].
\]
The first term belongs to $L^{p'}(\Omega)$ due to \eqref{frustone}.
For the second one this follows from a direct computation. This proves the claim and the proposition.
\end{proof}

\begin{definition}
\label{defcomp}
Let $u\in \widetilde{D}^{s,p}(\Omega)$ and $\Lambda\in (D^{s,p}_0(\Omega))^*$. We say that $(-\Delta_p)^s u\leq \Lambda$ weakly in $\Omega$ if for all $\varphi\in D^{s,p}_0(\Omega)$, $\varphi\ge 0$ in $\Omega$,
\[
\int_{\mathbb{R}^{2N}}\frac{J_p(u(x)-u(y))\,(\varphi(x)-\varphi(y))}{|x-y|^{N+s\,p}}\,dx\,dy \le \langle\Lambda, \varphi\rangle.
\]
\end{definition}

\begin{theorem}[Comparison principle in general domains]
\label{comparison}
Let $\Omega\subset\mathbb{R}^N$ be an open set. Let $u, v\in \widetilde{D}^{s,p}(\Omega)$ satisfy
\[
u\le v\ \mbox{ in }\Omega^c\qquad \mbox{ and }\qquad (-\Delta_p)^s u\leq (-\Delta_p)^s v\ \mbox{ in } \Omega.
\]
Then $u\le v$ in $\Omega$.
\end{theorem}

\begin{proof}
It suffices to proceed as in \cite[Lemma 9]{LL}, we only need to prove that $w:=(u-v)_+$ is an admissible test function, i.e. it belongs to $D^{s,p}_0(\Omega)$. Clearly $w\equiv 0$ in $ \Omega^c$ and $w\in L^{p^*}(\mathbb{R}^N)$. To estimate the Gagliardo seminorm, let $E\supset \Omega$ be such that $E^c$ is compact,  ${\rm dist}(E^c, \Omega)>0$ and
\begin{equation}
\label{kl}
\int_{E\times E} \frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\, dx\, dy+\int_{E\times E} \frac{|v(x)-v(y)|^p}{|x-y|^{N+s\,p}}\, dx\, dy<+\infty.
\end{equation}
Then
\[
\int_{\mathbb{R}^{2N}} \frac{|w(x)-w(y)|^p}{|x-y|^{N+s\,p}}\, dx\, dy=\int_{E\times E} \frac{|w(x)-w(y)|^p}{|x-y|^{N+s\,p}}\, dx\, dy+2\int_{\Omega\times E^c} \frac{|w(x)|^p}{|x-y|^{N+s\,p}}\, dx\, dy,
\]
and the first integral is finite due to
\[
|w(x)-w(y)|^p\leq C\,(|u(x)-u(y)|^p+|v(x)-v(y)|^p),
\]
and \eqref{kl}. For the second one we use \eqref{xmy}, and since $|w(x)|^p\leq C(|u(x)|^p+|v(x)|^p)$ we get
\[
\int_{\Omega\times E^c} \frac{|w(x)|^p}{|x-y|^{N+sp}}\, dx\, dy\leq C\,|E^c| \int_{\Omega} \frac{|u(x)|^p}{(1+|x|)^{N+sp}}\, dx+ C\,|E^c|\int_{\Omega} \frac{|v(x)|^p}{(1+|x|)^{N+sp}}\, dx.
\]
The last two terms are finite, due the definition of $\widetilde{D}^{s,p}(\Omega)$ and \eqref{frustone}.
\end{proof}

\noindent
Finally, for the reader's convenience we recall the following result from \cite{IMS}. The proof is identical to the one of \cite[Lemma 2.8]{IMS} and we omit it.

\begin{proposition}[Non-local behavior of $(-\Delta_p)^s$]
\label{nonlocalb}
Let $N>s\,p$ and let $\Omega\subset\mathbb{R}^N$ be an open set such that $\partial\Omega$ is compact and locally the graph of continuous functions. Suppose that $u\in \widetilde{D}^{s,p}(\Omega)$ weakly solves $(-\Delta_p)^su=f$  for some $f\in L^1_{\rm loc}(\Omega)\cap (D^{s,p}_0(\Omega))^*$, in the sense that
\begin{equation}
\label{defweaksol}
\langle (-\Delta_p)^s u, \varphi\rangle=\int_{\Omega}f\,\varphi\, dx,\qquad \mbox{ for every } \varphi\in D^{s,p}_0(\Omega).
\end{equation}
 Let $v$ be a measurable function with compact support $K:=\mathrm{supp}(v)$ such that
\[
{\rm dist}(K,\Omega)>0,\qquad \int_{\Omega^c} |v|^{p-1}\, dx<+\infty,
\]
and define for a.e.\ Lebesgue point $x\in \Omega$ of $u$
\[
h(x)=2\int_{K}\frac{J_p\big(\big(u(x)-u(y)\big)-v(y)\big)-J_p\big(u(x)-u(y)\big)}{|x-y|^{N+s\,p}}\,dy.
\]
Then $u+v\in \widetilde{D}^{s,p}(\Omega)$ and $(-\Delta_p)^s(u+v)=f+h$ weakly.
\end{proposition}

\subsection{Radial functions}

For every measurable function $u:\mathbb{R}^N\to\mathbb{R}$ we define its distribution function
\[
\mu_u(t)=\big|\{x\, :\, |u(x)|>t\}\big|,\qquad t>0.
\]
Let $0< q<\infty$ and $0<\theta<\infty$, the {\it Lorentz space} $L^{q,\theta}(\mathbb{R}^N)$ is defined by
\[
L^{q,\theta}(\mathbb{R}^N)=\left\{u\, :\, \int_0^\infty t^{\theta-1}\,\mu_u(t)^\frac{\theta}{q}\,dt<+\infty\right\}.
\]
In the limit case $\theta=\infty$, this is defined by
\[
L^{q,\infty}(\mathbb{R}^N)=\left\{u\, :\, \sup_{t>0} t\,\mu_u(t)^\frac{1}{q}<+\infty\right\},
\]
and we recall that this coincides with the weak $L^q$ space (see for example \cite[page 106]{liebL}).
\begin{lemma}[Radial Lemma for Lorentz spaces]
\label{lm:border}
Let $0<\theta\le \infty$ and $0< q<\infty$. Let $u\in L^{q,\theta}(\mathbb{R}^N)$ be a non-negative and radially symmetric decreasing function.
Then
\[
\begin{split}
0\le u(x)&\le \left(\theta\,\omega_N^{-\frac{\theta}{q}}\,\int_0^\infty t^{\theta-1}\,\mu_u(t)^\frac{\theta}{q}\,dt\right)^\frac{1}{\theta}\, |x|^{-\frac{N}{q}},\qquad \mbox{ if } \theta<\infty,\\
0\le u(x)&\le \left(\omega_N^{-\frac{1}{q}}\,\sup_{t>0} t\,\mu_u(t)^\frac{1}{q}\right)\,|x|^{-\frac{N}{q}},\qquad \mbox{ if } \theta=\infty.
\end{split}
\]
\end{lemma}
\begin{proof}
For simplicity, we simply write $\mu$ in place of $\mu_u$ and suppose that $u=u(r)$ coincides with its right-continuous representative. We start with the case $\theta<\infty$.
First of all, we prove that
\begin{equation}
\label{equivalente}
\int_0^\infty t^{\theta-1}\,\mu(t)^\frac{\theta}{q}\,dt=\frac{N-\alpha}{N\,\theta\,\omega_N^{\alpha/N}}\,\int_{\mathbb{R}^N} \frac{u^\theta}{|x|^\alpha}\,dx,
\end{equation}
where the exponent $\alpha<N$ is given by the relation\footnote{Observe that if $\theta>q$, then $\alpha<0$.}
\[
\frac{\theta}{q}=\frac{N-\alpha}{N}.
\]
With a simple change of variable
\begin{equation}
\label{cambio}
\int_0^\infty t^{\theta-1}\,\mu(t)^\frac{\theta}{q}\, dt=\frac{1}{\theta}\, \int_0^\infty \mu(s^{1/\theta})^\frac{\theta}{q}\, ds.
\end{equation}
Then we observe that
\[
\int_{\mathbb{R}^N} \frac{u^\theta}{|x|^\alpha}\, dx=\int_{\mathbb{R}^N} \frac{\displaystyle\int_0^\infty \chi_{\{t\,:\,u(x)^\theta>t\}}(s)\, ds}{|x|^\alpha}\, dx=\int_0^\infty \int_{\mathbb{R}^N} \frac{\chi_{\{t\,:\,u(x)>t^{1/\theta}\}}(s)}{|x|^\alpha}\, dx\, ds,
\]
and
\[
\chi_{\{t\,:\,u(x)>t^{1/\theta}\}}(s)=\chi_{\{y\, :\, u(y)>s^{1/\theta}\}}(x).
\]
By assumption we have
\[
\{y\, :\, u(y)>s^{1/\theta}\}=\left\{y\, :\, |y|<\left(\frac{\mu(s^{1/\theta})}{\omega_N}\right)^\frac{1}{N}\right\}=:B_{R(s)},
\]
since the function $u$ is radially decreasing. Thus we arrive at
\[
\begin{split}
\int_{\mathbb{R}^N} \frac{u^\theta}{|x|^\alpha}\, dx&=\int_0^\infty \left(\int_{B_{R(s)}} \frac{1}{|x|^\alpha}\, dx\right)\, ds\\
&=N\,\omega_N\, \int_0^\infty \int_0^{R(s)} \varrho^{N-1-\alpha}\,d\varrho\, ds=\frac{N\,\omega_N}{N-\alpha}\, \int_0^\infty \frac{\mu(s^{1/\theta})^\frac{N-\alpha}{N}}{\omega_N^\frac{N-\alpha}{N}}\, ds\\
&=\frac{N\,\omega_N^{\alpha/N}}{N-\alpha}\, \int_0^\infty \mu(s^{1/\theta})^\frac{\theta}{q}\, ds.
\end{split}
\]
Using \eqref{cambio} we finally obtain
\[
\int_0^\infty t^{\theta-1}\,\mu(t)^\frac{\theta}{q}\, dt=\frac{N-\alpha}{N\,\theta\,\omega_N^{\alpha/N}}\, \int_{\mathbb{R}^N} \frac{u^\theta}{|x|^\alpha}\, dx,
\]
which proves \eqref{equivalente}.
\vskip.2cm\noindent
As for the decay estimate, thanks to \eqref{equivalente} we have
\[
\begin{split}
+\infty>\int_{\mathbb{R}^N} \frac{u^\theta}{|x|^\alpha}\,dx&=N\,\omega_N\, \int_0^{+\infty} u(\varrho)^\theta\,\varrho^{N-1-\alpha}\, d\varrho\\
&\ge N\,\omega_N\, \int_0^{R} u(\varrho)^\theta\, \varrho^{N-1-\alpha}\,d\varrho\ge N\,\omega_N\, u(R)^\theta\, \frac{R^{N-\alpha}}{N-\alpha},
\end{split}
\]
where $\alpha$ is as above. Recalling that $(N-\alpha)/\theta=N/q$, we get the desired conclusion.
\vskip.2cm\noindent
For the case $\theta=\infty$, it is sufficient to observe that
\[
\sup_{t>0} t\,\mu(t)^\frac{1}{q}=\omega_N^\frac{1}{q}\,\sup_{x\in\mathbb{R}^N} |x|^\frac{N}{q}\,u(x).
\]
Then the decay estimate easily follows.
\end{proof}

\section{Properties of extremals}

\label{sec:3}

\subsection{Basic properties}
We first observe that by homogeneity we can equivalently write
\begin{equation}
\label{costante}
\mathcal{S}_{p,s}=\inf_{u\in D^{s,p}(\mathbb{R}^N)} \left\{\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\,p}}\,dx\,dy\, :\, \int_{\mathbb{R}^N} |u|^{{N\,p/(N-s\,p)}}\,dx=1\right\}.
\end{equation}
Then we start with the following result.
\begin{proposition}
\label{prop:4cazzate}
Let $1<p<\infty$ and $s\in(0,1)$ be such that $s\,p<N$. Then:
\begin{itemize}
\item problem \eqref{costante} admits a solution;
\vskip.2cm
\item for every $U\in D^{s,p}(\mathbb{R}^N)$ solving \eqref{costante}, there exist $x_0\in\mathbb{R}^N$ and $u:\mathbb{R}^+\to\mathbb{R}$ constant sign monotone function such that $U(x)=u(|x-x_0|)$;
\vskip.2cm
\item every minimizer $U\in D^{s,p}(\mathbb{R}^N)$ weakly solves
\[
(-\Delta_p)^s U=\mathcal{S}_{p,s}\,|U|^{p^*-2}\,U,\qquad \mbox{ in }\mathbb{R}^N,
\]
that is
\begin{equation}
\label{eulero}
\int_{\mathbb{R}^{2N}}  \frac{J_p(U(x)-U(y))\, \big(\varphi(x)-\varphi(y)\big)}{|x-y|^{N+s\,p}}\,dx\,dy=\mathcal{S}_{p,s}\, \int_{\mathbb{R}^N} |U|^{p^*-2}\,U\,\varphi\,dx,
\end{equation}
for every $\varphi\in D^{s,p}(\mathbb{R}^N)$.
\end{itemize}
\end{proposition}
\begin{proof}
The existence of a solution for \eqref{costante} follows from the Concentration-Compactness Principle, see \cite[Section I.4, Example iii)]{Lions}.
\par
It is not difficult to show that every solution of \eqref{costante} must have costant sign. Indeed, for every admissible $u\in D^{s,p}(\mathbb{R}^N)$, still by \eqref{valorassoluto} the function $|u|$ is still admissible and does not increase the value of the functional.
More important, the inequality sign in \eqref{valorassoluto} is strict if $u(x)\,u(y)<0$, i.e. if $u$ changes sign.
\par
Radial symmetry of the solutions comes from the {\it P\'olya-Szeg\H{o} principle} for Gagliardo seminorms (see \cite{AlLi}), i.e. for every non-negative function $u\in D^{s,p}(\mathbb{R}^N)$ we have
\begin{equation}
\label{sp}
[u^\#]^p_{s,p}\le [u]^p_{s,p}.
\end{equation}
Here $u^\#$ denotes the radially symmetric decreasing rearrangement of $u$. It is crucial to observe that inequality \eqref{sp} is strict, unless $u$ is (up to a translation) a radially symmetric decreasing function, see \cite[Theorem A.1]{FS}.
\par
Finally, if $U$ solves \eqref{costante}, then it minimizes as well the functional
\[
u\mapsto [u]^p_{s,p}-\mathcal{S}_{p,s}\,\left(\int_{\mathbb{R}^N} |u|^{p^*}\,dx\right)^\frac{p}{p^*}.
\]
Equation \eqref{eulero} is exactly the Euler-Lagrange equation associated with this functional, once it is observed that $U$ has unitary $L^{p^*}$ norm.
\end{proof}
\begin{proposition}[Global boundedness]
Let $U\in D^{s,p}(\mathbb{R}^N)$ be a non-negative solution of \eqref{costante}. Then we have $U\in L^\infty(\mathbb{R}^N)\cap C^0(\mathbb{R}^N)$.
\end{proposition}
\begin{proof}
Thanks to the properties of the minimizers contained in Proposition \ref{prop:4cazzate}, it is  enough to prove that $U\in L^\infty_{\mathrm{\mathrm{loc}}}(\mathbb{R}^N)$, since continuity then follows from \cite[Theorem 3.13]{BP} (see also \cite[Theorem 5.4]{IMS} for a direct proof).
With this aim, we just need to show that $U\in L^{q\,(p^*-1)}(\mathbb{R}^N)$ for some $q>N/(s\,p)$. This would imply that
\[
U^{p^*-1}\in L^q(\mathbb{R}^N),\qquad \mbox{ for some }q>\frac{N}{s\,p},
\]
and thus $U\in L^\infty_{\mathrm{loc}}(\mathbb{R}^N)$ would automatically follow by \cite[Theorem 3.8]{BP}.
\vskip.2cm\noindent
Let $M>0$ and $\alpha>1$, we set for simplicity $U_M=\min\{U,M\}$ and $g_{\alpha,M}(t)=t\,\min\{t,\,M\}^{\alpha-1}$. Then we insert in \eqref{eulero} the test function $\varphi=g_{\alpha,M}(U)\in D^{s,p}(\mathbb{R}^N)$. This yields
\[
\begin{split}
\int_{\mathbb{R}^{2N}} & \frac{J_p(U(x)-U(y))\, \big(g_{\alpha,M}(U(x))-g_{\alpha,M}(U(y))\big)}{|x-y|^{N+s\,p}}\,dx\,dy=\mathcal{S}_{p,s}\, \int_{\mathbb{R}^N} U^{p^*-p}\,U_{M}^{\alpha-1}\,U^p\,dx.\\
\end{split}
\]
We now observe that if we set
\[
G_{\alpha,M}(t)=\int_0^t g'_{\alpha,M}(\tau)^\frac{1}{p}\,d\tau,
\]
by using \cite[Lemma A.2]{BP} from the previous identity with simple manipulations we get
\[
\begin{split}
\int_{\mathbb{R}^{2N}} &\frac{|G_{\alpha,M}(U(x))-G_{\alpha,M}(U(y))|^p}{|x-y|^{N+s\,p}}\,dx\,dy\\
&\le \mathcal{S}_{p,s}\, \left[K_0^{\alpha-1}\,\int_{\mathbb{R}^N} U^{p^*}\,dx+\left(\int_{\{U\ge K_0\}} U^{p^*}\,dx\right)^\frac{p^*-p}{p^*}\,\left(\int_{\mathbb{R}^N} \left(U_{M}^{(\alpha-1)}\,U^{p}\right)^\frac{p^*}{p}\,dx\right)^\frac{p}{p^*}\right],
\end{split}
\]
for some $K_0>0$ that will be chosen in a while.
If we estimate from below the left-hand side by Sobolev inequality and use that $U$ has unitary norm, we get\footnote{Here we use that
\[
G_{\alpha,M}(t)\ge \frac{p}{p+\alpha-1}\, t\,\min\{t,\,M\}^\frac{\alpha-1}{p}.
\]}
\begin{equation}
\label{intermediate}
\begin{split}
\left(\frac{p}{p+\alpha-1}\right)^p\,\left(\int_{\mathbb{R}^N} \right.&\left.\left(U^p\,U_M^{(\alpha-1)}\right)^\frac{p^*}{p}\,dx\right)^\frac{p}{p^*}\le K_0^{\alpha-1}\, \\
&+\left(\int_{\{U\ge K_0\}} U^{p^*}\,dx\right)^\frac{p^*-p}{p^*}\,\left(\int_{\mathbb{R}^N} \left(U_M^{(\alpha-1)}\,U^{p}\right)^\frac{p^*}{p}\,dx\right)^\frac{p}{p^*}.
\end{split}
\end{equation}
We now choose the parameters: we first take $\alpha>1$ such that
\[
p^*+(\alpha-1)\,\frac{p^*}{p}=q\,(p^*-1),\qquad \mbox{ i.\,e. } \quad \alpha=p\,q\,\frac{(p^*-1)}{p^*}-(p-1),
\]
where $q>N/(s\,p)$,
then we choose $K_0=K_0(\alpha,U)>0$ such that
\[
\left(\int_{\{U\ge K_0\}} U^{p^*}\,dx\right)^\frac{p^*-p}{p^*}\le \frac{1}{2}\,\left(\frac{p}{p+\alpha-1}\right)^p.
\]
With this choice we can absorb the last term on the right-hand side of \eqref{intermediate} and thus obtain
\[
\left(\frac{p}{p+\alpha-1}\right)^p\,	\left(\int_{\mathbb{R}^N} U^{p^*}\,U_M^{(\alpha-1)\,\frac{p^*}{p}}\,dx\right)^\frac{p}{p^*}\le 2\,K_0^{\alpha-1}.
\]
If we now take the limit as $M$ goes to $+\infty$, we finally get that $U\in L^{q\,(p^*-1)}(\mathbb{R}^N)$ for some $q>N/(s\,p)$, together with the estimate
\[
\left\|U^{p^*-1}\right\|^q_{q}\le \left(2\,K_0^{\alpha-1}\,\left(\frac{p+\alpha-1}{p}\right)^p\right)^\frac{p^*}{p},
\]
and thus the conclusion.
\end{proof}

\begin{proposition}[Borderline Lorentz estimate]
\label{lp-weak}
Let $U\in D^{s,p}(\mathbb{R}^N)$ be a non-negative solution of \eqref{costante}. Then
\begin{equation}
\label{cazzata}
U\in L^q(\mathbb{R}^N),\qquad \mbox{ for every } q>q_0:=\frac{(p-1)\,N}{N-s\,p}.
\end{equation}
Moreover, we have $U\in L^{q_0,\infty}(\mathbb{R}^N)$ with the estimate
\begin{equation}
\label{stimalpdebole}
\sup_{t>0}t\,|\{U> t\}|^\frac{1}{q_0}\leq \|U\|_{p^*-1}^{\frac{p^*-1}{p-1}}.
\end{equation}
\end{proposition}
\begin{proof}
We divide the proof in two parts: we first prove \eqref{cazzata}.
Then we will use \eqref{cazzata} to prove \eqref{stimalpdebole}.
\vskip.2cm\noindent
{\it Part I: intermediate estimate}.
Given $0<\alpha<1$ and
$\varepsilon>0$, we take the Lipschitz increasing function $\psi_\varepsilon:[0,+\infty)\to[0,+\infty)$
defined as
\[
\psi_\varepsilon(t)=\int_0^t \left[(\varepsilon+\tau)^\frac{\alpha-1}{p}+\frac{\alpha-1}{p}\,\tau\,(\varepsilon+\tau)^\frac{\alpha-1-p}{p}\right]^p\,d\,\tau.
\]
We observe that
\begin{equation}
\label{geps}
0\le \psi_\varepsilon(t)\le \int_0^t (\varepsilon+t)^{\alpha-1}\,d\tau=\frac{1}{\alpha}\,[(\varepsilon+t)^\alpha-\varepsilon^\alpha]\le \frac{t^\alpha}{\alpha},
\end{equation}
where in the second inequality we used that $0<\alpha<1$.
We insert in \eqref{eulero} the test function $\varphi=\psi_\varepsilon(U)\in D^{s,p}(\mathbb{R}^N)$. This gives
\[
\begin{split}
\int_{\mathbb{R}^{2N}} & \frac{J_p(U(x)-U(y))\, \big(\psi_{\varepsilon}(U(x))-\psi_\varepsilon(U(y)\big)}{|x-y|^{N+s\,p}}\,dx\,dy=\mathcal{S}_{p,s}\, \int_{\mathbb{R}^N} U^{p^*-1}\,\psi_\varepsilon(U)\,dx.\\
\end{split}
\]
By defining
\[
\Psi_\varepsilon(t):=\int_0^t \psi'_\varepsilon(\tau)^\frac{1}{p}\,d\tau	=t\,(\varepsilon+t)^\frac{\alpha-1}{p},
\]
if we proceed as in the previous proof and use \eqref{geps}, we get
\[
\begin{split}
\left(\int_{\mathbb{R}^N} \Psi_\varepsilon(U)^{p^*}\,dx\right)^\frac{p}{p^*}&\le \frac{1}{\alpha}\,\|U\|^{p^*+\alpha-1}_{\infty}\,|\{U>K_0\}|\\
&\quad +\left(\int_{\{U\le K_0\}} U^{p^*}\,dx\right)^\frac{p^*-p}{p^*}\left(\int_{\{U\le K_0\}} \Big(\psi_\varepsilon(U)\,U^{p-1}\Big)^\frac{p^*}{p}\,dx\right)^\frac{p}{p^*},
\end{split}
\]
for $K_0>0$. Observe that we also used the previous Proposition to assure that $U\in L^\infty(\mathbb{R}^N)$.
From \eqref{geps} we get
\begin{equation*}
0\le \psi_\varepsilon(t)\,t^{p-1}\le \frac{1}{\alpha}\,[(\varepsilon+t)^\alpha-\varepsilon^\alpha]\,t^{p-1}
\le \quad \frac{1}{\alpha}\,(\varepsilon+t)^{\alpha-1}\,t^{p}=\frac{1}{\alpha}\, \Psi_\varepsilon(t)^p.
\end{equation*}
Thus we arrive at
\[
\begin{split}
\left(\int_{\mathbb{R}^N} \Psi_\varepsilon(U)^{p^*}\,dx\right)^\frac{p}{p^*}&\le \frac{1}{\alpha}\,\|U\|^{p^*+\alpha-1}_{\infty}\,|\{U>K_0\}|\\
&+\frac{1}{\alpha}\,\left(\int_{\{U\le K_0\}} U^{p^*}\,dx\right)^\frac{p^*-p}{p^*}\,\left(\int_{\mathbb{R}^N} \Psi_\varepsilon(U)^{p^*}\,dx\right)^\frac{p}{p^*},
\end{split}
\]
The level $K_0=K_0(\alpha,U)>0$ is now chosen so that
\[
\left(\int_{\{U\le K_0\}} U^{p^*}\,dx\right)^\frac{p^*-p}{p^*}\le \frac{\alpha}{2},
\]
which yields
\[
\left(\int_{\mathbb{R}^N} \left(U\,(U+\varepsilon)^\frac{\alpha-1}{p}\right)^{p^*}\,dx\right)^\frac{p}{p^*}\le \frac{2}{\alpha}\,\|U\|^{p^*+\alpha-1}_{\infty}\,|\{U>K_0\}|,
\]
for every $0<\alpha<1$. By taking the limit as $\varepsilon$ goes to $0$, we get the desired integrability \eqref{cazzata}.
\vskip.2cm\noindent
{\it Part II: borderline Lorentz estimate.} We now prove \eqref{stimalpdebole}.
For any $t>0$ we let $g_t(s)=\min\{t,\,s\}$, and define
\[
G_t(s)=\int_0^sg'_t(\tau)^{\frac 1 p}d\tau=g_t(s).
\]
We test \eqref{eulero} with $g_t(U)$ and, thanks to \cite[Lemma A.2]{BP} and Sobolev inequality we get
\[
\begin{split}
\mathcal{S}_{p,s}\,\|g_t(U)\|_{p^*}^p&\leq [g_t(U)]_{s, p}^p\leq \int_{\mathbb{R}^{2N}} \frac{J_p(U(x)-U(y))\, \big(g_t(U(x))-g_t(U(y))\big)}{|x-y|^{N+s\,p}}\,dx\,dy\\
&\leq \mathcal{S}_{p,s}\,\int_{\mathbb{R}^N} U^{p^*-1}\,g_t(U)\,dx.
\end{split}
\]
We have $U\in L^{p^*-1}(\mathbb{R}^N)$, by choosing  $q=p^*-1$ in \eqref{cazzata}. Thus we conclude that
\[
t\,|\{U>t\}|^{\frac{1}{p^*}}\leq \|g_t(U)\|_{p^*}\leq \left(\int_{\mathbb{R}^N} U^{p^*-1}g_t(U)\,dx\right)^\frac{1}{p}\leq t^\frac{1}{p}\,\|U\|_{p^*-1}^\frac{p^*-1}{p}.
\]
This finally yields \eqref{stimalpdebole}, after some elementary manipulations.
\end{proof}

\subsection{Decay estimates}

As an intermediate step towards the proof of the asymptotic result \eqref{sciarpa2}, in this subsection we will prove that any (positive) solution of \eqref{costante} verifies
\[
\frac{1}{C}\,|x|^{-\frac{N-s\,p}{p-1}}\le U(x)\le C\,|x|^{-\frac{N-s\,p}{p-1}},\qquad |x|> 1,
\]
for some $C=C(N,p,s,U)>1$,
see Corollary \ref{lower-Dec} below.
\vskip.2cm\noindent
In what follows, we will set for simplicity
\[
\Gamma(x)=|x|^{-\frac{N-s\,p}{p-1}},\qquad x\in\mathbb{R}^N\setminus\{0\},
\]
and
\begin{equation}
\label{stroncata}
\widetilde \Gamma(x)=\min\{1,\,\Gamma(x)\}=\min\left\{1,\, |x|^{-\frac{N-s\,p}{p-1}}\right\},\qquad x\in\mathbb{R}^N.
\end{equation}
The following expedient result will be useful.
\begin{lemma}
\label{lm:stroncalaplaciani!}
With the notation above, we have
\begin{equation}
\label{splv}
\frac{1}{C}\,\,|x|^{-N-s\,p}\leq (-\Delta_p)^s \widetilde\Gamma(x)\leq C\,|x|^{-N-s\,p},\qquad  \mbox{ for }|x|>R>1,
\end{equation}
in weak sense, for some $C=C(N,p,s,R)>1$. The constant blows-up as $R$ goes to $1$.
\end{lemma}
\begin{proof}
From Theorem \ref{fundamental}, we know that $\Gamma$ belongs to $\widetilde{D}^{s,p}(B_R^c)$ and
is a weak solution of $(-\Delta_p)^su=0$ in $B_R^c$ for any $R>1$. We then observe that the truncated function $\widetilde \Gamma$ can be written as
\[
\widetilde \Gamma(x)=\Gamma(x)-\big(\Gamma(x)-1\big)_+.
\]
Thus we apply Proposition \ref{nonlocalb}, with the choices
\[
\Omega=B_R^c,\qquad u=\Gamma,\qquad f\equiv 0,\qquad v=-(\Gamma-1)_+,
\]
This yields for $|x|>R$
\begin{equation}
\label{stroncalaplaciano}
\begin{split}
(-\Delta_p)^s \widetilde\Gamma (x)&=2\,\int_{B_1}\frac{J_p(\Gamma(x)-1)-J_p(\Gamma(x)-\Gamma(y))}{|x-y|^{N+s\,p}}\, dy\\
&=2\,\int_{B_1}\frac{J_p(\Gamma(y)-\Gamma(x))-J_p(1-\Gamma(x))}{|x-y|^{N+s\,p}}\, dy.
\end{split}
\end{equation}
We first prove the upper bound in \eqref{splv}. To this aim, by the monotonicity of $\Gamma$ we get
\[
(\Gamma(y)-\Gamma(x))^{p-1}-(1-\Gamma(x))^{p-1}\leq (\Gamma(y)-\Gamma(x))^{p-1}\leq \Gamma(y)^{p-1},\qquad |x|> R, \ |y|\leq 1.
\]
Moreover
\[
|x-y|\geq \frac{R-1}{R}\,|x|,\qquad \mbox{ for all } |x|>R \mbox{ and } |y|<1.
\]
By spending these informations in \eqref{stroncalaplaciano}, we obtain
\[
(-\Delta_p)^s \widetilde\Gamma (x)\leq \left(\frac{R}{R-1}\right)^{N+s\,p}\,\frac{2}{|x|^{N+s\,p}}\int_{B_1} \Gamma(y)^{p-1}\, dy=\frac{C}{|x|^{N+s\,p}},
\]
as desired.
Observe that we also used that $\Gamma\in L^{p-1}_{\mathrm{loc}}(\mathbb{R}^N)$.
\vskip.2cm\noindent
In order to prove the lower bound, we need to distinguish between the case $1<p<2$ and the case $p\ge 2$. If $p\ge 2$, then $J_p(t)=|t|^{p-2}\,t$ is a convex superadditive function on $[0,\infty)$. Thus we get
\[
J_p(\Gamma(y)-\Gamma(x))-J_p(1-\Gamma(x))\ge J_p(\Gamma(y)-1),\qquad |x|>R>1>|y|.
\]
As for the kernel, we have
\begin{equation}
\label{boffo}
|x-y|<2\,|x|,\qquad \mbox{ if }|x|>|y|,
\end{equation}
thus in conclusion from \eqref{stroncalaplaciano} we get
\[
(-\Delta_p)^s \widetilde\Gamma (x)\ge \frac{2^{1-N-s\,p}}{|x|^{N+s\,p}}\int_{B_1} \big(\Gamma(y)-1\big)^{p-1}\, dy=\frac{C}{|x|^{N+s\,p}}.
\]
Using again that $\Gamma\in L^{p-1}_{\mathrm{loc}}(\mathbb{R}^N)$ and that $\Gamma>1$ in $B_1$ gives the lower bound in \eqref{splv}, in the case $p\ge 2$.
\par
In the case $1<p<2$, we need to use \eqref{seconda}, which gives
\[
\begin{split}
J_p(\Gamma(y)-\Gamma(x))-J_p(1-\Gamma(x))&\ge C\,\frac{(\Gamma(y)-1)}{\Big((\Gamma(y)-\Gamma(x))^2+(1-\Gamma(x))^2\Big)^\frac{2-p}{2}}\\
&\ge \frac{C}{2^\frac{2-p}{2}}\,\frac{(\Gamma(y)-1)}{(\Gamma(y)-\Gamma(x))^{2-p}}\ge C\,\Gamma(y)^{p-1}\, \left(1-\frac{1}{\Gamma(y)}\right).
\end{split}
\]
By using this and \eqref{boffo} in \eqref{stroncalaplaciano}, we get the desired lower bound for $1<p<2$ as well.
\end{proof}
\noindent
In order to prove a lower bound for positive radially decreasing solutions of \eqref{costante}, we need to focus on the auxiliary problem
\begin{equation}
\label{cappot}
\mathcal{I}(R)=\inf_{u\in D^{s,p}(\mathbb{R}^N)}\Big\{[u]_{s, p}^p: u\geq \chi_{B_R}\Big\}.
\end{equation}
\begin{proposition}
Let $1<p<\infty$ and $s\in(0,1)$ be such that $s\,p<N$. For any $R>0$, problem \eqref{cappot} has a unique solution $u_R>0$. Moreover, $u_R$ is radial, non-increasing and  $u_R\in D^{s,p}(\mathbb{R}^N)$ solves in weak sense
\[
\left\{\begin{array}{rcll}
(-\Delta_p)^s u_R&=&0&\text{ in }  \overline{B_R}^{\,c},\\
u_R&\equiv& 1&\mbox{ in } \overline{B_R}.
\end{array}
\right.
\]
\end{proposition}
\begin{proof}
The existence of a solution follows easily by using the Direct Methods. Indeed, if $\{u_n\}_{n\in\mathbb{N}}\subset D^{s,p}(\mathbb{R}^N)$ is a minimizing sequence, then a uniform bound on their Gagliardo seminorms entails a uniform bound on the $L^{p^*}$ norms, by Sobolev inequality. Thus we have weak convergence (up to a subsequence) in $L^{p^*}(\mathbb{R}^N)$ to a function $u\in D^{s,p}(\mathbb{R}^N)$. Moreover, the constraint $u_n\ge \chi_{B_R}$ is stable with respect to weak convergence and thus it passes to the limit. Consequently, $u$ is a minimizer. The uniqueness follows from strict convexity of the functional.
\par
All the other required properties of $u_R$ follow as in the proof of Proposition \ref{prop:4cazzate}, we just show that $u_R$ saturates the constraint $u_R\ge \chi_{B_R}$. For simplicity, we set $\mathcal{E}(u)=[u]_{s,p}^p$. Then from \cite[Remark 3.3]{GM} we have
\begin{equation}
\label{submodular}
\mathcal{E}(\max\{u,\, t\})+\mathcal{E}(\min\{u,\, t\})\leq \mathcal{E}(u),\qquad \mbox{ for every } u\in D^{s,p}(\mathbb{R}^N),\ t\in \mathbb{R}.
\end{equation}
In particular, $\min\{u_R,1\}$ is admissible and is still a minimizer. Thus by uniqueness it coincides with $u_R$.
\end{proof}
Thanks to Lemma \ref{lm:stroncalaplaciani!}, we can prove a decay estimate for the solution of \eqref{cappot}.
\begin{proposition}
\label{prop:potenziale}
The solution $u_1$ of problem \eqref{cappot} with $R=1$ satisfies
\[
\frac{|x|^{-\frac{N-s\,p}{p-1}}}{C}\leq u_1(x)\leq p^\frac{1}{p-1}\,|x|^{-\frac{N-s\,p}{p-1}},\qquad \mbox{ for } |x|\geq 1,
\]
for some constant $C=C(N,p,s)>1$.
\end{proposition}

\begin{proof} Observe that $u_1$ is continuous due to \cite[Theorem 1.1]{IMS}. We prove the two estimates separately.
\vskip.2cm\noindent
{\em Upper bound.}
We first observe that by using the scaling properties of the Gagliardo seminorm, we have
\begin{equation}
\label{scaling}
\mathcal{I}(R)=R^{N-s\,p}\,\mathcal{I}(1).
\end{equation}For every $R>1$, we set $u_1(R)=t \in (0,1)$. As in the previous proof, we set $\mathcal{E}(u)=[u]_{s,p}^p$.
The function $\min\{u_1,\, t\}/t$ is admissible for problem \eqref{cappot} with $B_R$, then the minimality of $u_R$ gives
\[
\mathcal{E}\left(\frac{\min\{u_1,\,t\}}{t}\right)\geq \mathcal{E}(u_R)=\mathcal{I}(R)=R^{N-s\,p}\,\mathcal{I}(1),
\]
thanks to \eqref{scaling}.
Similarly, we get
\[
\mathcal{E}\left(\frac{\max\{u_1-t,\, 0\}}{1-t}\right)\geq \mathcal{E}(u_1)=\mathcal{I}(1).
\]
then using the $p-$homogeneity of the energy and summing the previous two inequalities
\[
\mathcal{E}(\min\{u_1,\, t\})+\mathcal{E}(\max\{u_1,\, t\})\geq \big(t^p\,R^{N-s\,p}+(1-t)^p\big)\,\mathcal{I}(1).
\]
Using the submodularity of Gagliardo seminorms \eqref{submodular} in the left-hand side and simplifying we get
\[
t^p\,R^{N-s\,p}\leq 1-(1-t)^p.
\]
By recalling the definition of $t$, we obtain
\begin{equation}
\label{decu1}
u_1(R)^p\, R^{N-s\, p}\leq 1-(1-u_1(R))^p
\end{equation}
and since $1-(1-u_1(R))^p\leq p\, u_1(R)$ we get
\[
u_1(R)\leq p^{\frac{1}{p-1}}\, R^{-\frac{N-sp}{p-1}}.
\]
\vskip.2cm\noindent
{\em Lower bound}.
By using Proposition \ref{nonlocalb} with
\[
\Omega=\overline{B_3}^{\,c},\qquad u=u_1,\qquad f\equiv 0,\qquad v=-(u_1-u_1(2))_+,
\]
the truncated function
\[
u=\min\{u_1,\,u_1(2)\}=u_1-(u_1-u_1(2))_+,
\]
satisfies weakly in $\overline{B_3}^{\,c}$
\[
\begin{split}
(-\Delta_p)^s u(x)&=2\int_{B_2}\frac{J_p(u_1(x)-u_1(2))-J_p(u_1(x)-u_1(y))}{|x-y|^{N+s\,p}}\, dy\\
&\geq 2\int_{B_1}\frac{J_p(u_1(y)-u_1(x))-J_p(u_1(2)-u_1(x))}{|x-y|^{N+s\,p}}\, dy.
\end{split}
\]
In the last passage we used that the integrand is nonnegative by the monotonicity of $u_1$. Recall that $u_1\equiv 1$ in $B_1$ and by \eqref{decu1} we have $u_1(2)<1=u_1(1)$. Then, it is readily checked that
\[
(u_1(1)-u_1(x))^{p-1}-(u_1(2)-u_1(x))^{p-1}\geq c,\qquad  \mbox{ for }|x|>3,
\]
for some constant $c=c(p,u_1(1)-u_1(2))>0$.
Since also $|x-y|\leq 2\,|x|$ for all $x\in B_2^c$ and $y\in B_1$, the previous discussion yields
\begin{equation}
\label{stimau}
(-\Delta_p)^s u(x)\geq \frac{2\,c\,|B_1|}{(2\,|x|)^{N+s\,p}}=:\frac{c_1}{|x|^{N+s\,p}},\qquad \mbox{ for }|x|>3.
\end{equation}
On the other hand, from Lemma \ref{lm:stroncalaplaciani!}, for every $\varepsilon>0$ we have \begin{equation}
\label{stimavepsilon}
(-\Delta_p)^s (\varepsilon\,\widetilde\Gamma (x))\le \frac{c_2}{|x|^{N+s\,p}}\,\varepsilon^{p-1},\qquad \mbox{ for } |x|>3,
\end{equation}
where $\widetilde \Gamma$ is given in \eqref{stroncata}.
Now choose $\varepsilon>0$ as follows
\[
\varepsilon=\min\left\{u_1(3)\,3^\frac{N-s\,p}{p-1},\, \left(\frac{c_1}{c_2}\right)^\frac{1}{p-1}\right\},
\]
so that by \eqref{stimau} and \eqref{stimavepsilon} it holds
\[
\left\{\begin{array}{rcll}
(-\Delta_p)^s (\varepsilon\,\widetilde\Gamma)\!\!\! &\leq&\!\!\! (-\Delta_p)^s u&\text{ in } \overline{B_3}^{\,c},\\
\varepsilon\,\widetilde\Gamma\!\!\!&\leq&\!\!\! u &\text{ in } \overline{B_3}.
\end{array}
\right.
\]
Therefore by Theorem \ref{comparison} and the definitions of $\widetilde \Gamma$ and $u$ we have
\[
\varepsilon\,|x|^{-\frac{N-s\,p}{p-1}}\leq u(x)=u_1(x),\quad \mbox{ for }|x|>3.
\]
In $\overline{B_3}\setminus B_1$ the estimate is simpler to obtain, indeed
\[
u_1(3)\,|x|^{-\frac{N-s\,p}{p-1}}\le u_1(3)\le u_1(x),
\]
thus we get the conclusion.
\end{proof}
Finally, we can prove the aforementioned decay estimate for solutions of \eqref{costante}.
\begin{corollary}[Sharp decay rate]
\label{lower-Dec}
Let $U\in D^{s,p}(\mathbb{R}^N)$ be a positive radially symmetric
and decreasing solution of \eqref{costante}. Then
\[
\left(\inf_{B_1} U\right)\,\frac{|x|^{-\frac{N-s\,p}{p-1}}}{C}\le U(x)\le \left(\omega_N^{-\frac{1}{p^*}}\,\|U\|_{p^*-1}^{\frac{p^*-1}{p}}\right)^\frac{p}{p-1}\,|x|^{-\frac{N-s\,p}{p-1}},\qquad |x|\geq 1,
\]
where the constant $C=C(N,p,s)>1$ is the same of Proposition \ref{prop:potenziale}.
\end{corollary}

\begin{proof}
The upper bound follows from the borderline $L^{q_0,\infty}$ estimate of \eqref{stimalpdebole}, combined with the Radial Lemma \ref{lm:border}.
\par
As for the lower bound, by the weak Harnack inequality for positive supersolution of $(-\Delta_p)^s$ (see \cite[Theorem 5.2]{IMS}), we have
\[
\lambda:=\inf_{B_1} U\ge C\,\left(\int_{B_2} U^{p-1}\,dx\right)^\frac{1}{p-1}>0.
\]
Then the function $\lambda\, u_1$ is a lower barrier for $U$ in $B_1^c$. Thus the lower bound follows from Theorem \ref{comparison} and Proposition \ref{prop:potenziale}.
\end{proof}

\section{Proof of the main result}

\label{sec:4}
In this section we still denote by $\widetilde \Gamma$ the truncated function defined by \eqref{stroncata}, while $U$ is a positive radially symmetric
and decreasing solution of \eqref{costante}. As in the previous section we will systematically use the abuse of notation $U(x)=U(r)$ and $\widetilde \Gamma(x)=\widetilde \Gamma(r)$, for $r=|x|$.

\begin{lemma}
\label{lm:intervalli}
Suppose that
$$
U(R)\geq A\,\widetilde \Gamma(R)\qquad\text{for some $R>2$}.
$$
For any $\delta>0$ there exists $\theta=\theta(N, p, s, \delta,U)<1$ such that
\[
U(r)\geq (A-\delta)\,\widetilde\Gamma(r)\qquad \mbox{ for any }\theta\, R\leq r\leq R.
\]
Similarly, if
$$
U(R)\leq B\, \widetilde \Gamma(R)\qquad \text{for some $R>2$},
$$
then
\[
U(r)\leq (B+\delta)\, \widetilde\Gamma(r)\qquad \mbox{ for any }R\leq r\leq R/\theta.
\]
\end{lemma}

\begin{proof}
Consider the first statement and let $\theta<1$ to be determined. $U$ is non increasing and
$$
U(r)\leq C\,r^{-\frac{N-s\,p}{p-1}}, \qquad r\ge 1,
$$
by Corollary \ref{lower-Dec}.
Then for any $\theta\, R\leq r\leq R$ it holds
\[
\frac{U(R)}{\widetilde\Gamma(R)}-\frac{U(r)}{\widetilde\Gamma(r)}\leq U(R)\,\left(\frac{1}{\widetilde\Gamma(R)}-\frac{1}{\widetilde\Gamma(r)}\right)\leq \frac{C}{R^{\frac{N-s\,p}{p-1}}}\,\left(R^{\frac{N-s\,p}{p-1}}-r^{\frac{N-s\,p}{p-1}}\right)\leq C\,\left(1-\theta^{\frac{N-s\,p}{p-1}}\right).
\]
Therefore by hypothesis we get
\[
\frac{U(r)}{\widetilde\Gamma(r)}\geq A-C\,\left(1-\theta^{\frac{N-s\,p}{p-1}}\right),\qquad \mbox{ for }  r\in [\theta\, R, R],
\]
which gives the first claim. The proof of the other statement is similar: for any $R\leq r\leq R/\theta$ it holds
\[
\frac{U(r)}{\widetilde\Gamma(r)}-\frac{U(R)}{\widetilde\Gamma(R)}\leq U(R)\,\left(\frac{1}{\widetilde\Gamma(r)}-\frac{1}{\widetilde\Gamma(R)}\right)\leq  \frac{C}{R^{\frac{N-sp}{p-1}}}\,\left(r^{\frac{N-s\,p}{p-1}}-R^{\frac{N-s\,p}{p-1}}\right)\leq C\,\left(\theta^{-\frac{N-s\,p}{p-1}}-1\right),
\]
which gives
\[
\frac{U(r)}{\widetilde\Gamma(r)}\leq B+C\,\left(\theta^{-\frac{N-sp}{p-1}}-1\right),\qquad \mbox{ for } r\in [R, R/\theta].
\]
This completes the proof.
\end{proof}

\noindent
We are ready for the proof of the main result.
\begin{theorem}
There exists $U_\infty>0$ such that
$$
\lim_{r\to +\infty} r^{\frac{N-s\,p}{p-1}}U(r)=U_\infty.
$$
\end{theorem}
\begin{proof}
We can suppose that $p\neq 2$, since for $p=2$ the function $U$ has an explicit expression.
By virtue of Corollary \ref{lower-Dec} we readily have
\[
\frac{1}{C}\leq m:=\liminf_{r\to +\infty} \frac{U(r)}{\widetilde\Gamma(r)}\leq \limsup_{r\to +\infty}\frac{U(r)}{\widetilde\Gamma(r)}=:M\leq C,
\]
with $C$ depending on $U$ as well.
Suppose by contradiction that $M-m>0$, and fix $0<\varepsilon_0<(M-m)/4$.
\vskip.2cm\noindent
$\bullet$ \underline{\em Case $p>2$}.
There exists $R_0=R_0(\varepsilon_0)>2$ such that
\[
\frac{U(r)}{\widetilde\Gamma(r)}\geq m-\varepsilon_0,\quad \mbox{ for }r\geq R_0,
\]
and we can choose an arbitrarily large $R>R_0$ such that
$$
\frac{U(R)}{\widetilde\Gamma(R)}\geq M-\frac{M-m}{4}.
$$
Consider $\delta=(M-m)/4$. By Lemma \ref{lm:intervalli}, there exists $\theta<1$ so that for any such $R$ it holds
\begin{equation}
\label{uvgeq}
\frac{U(r)}{\widetilde\Gamma(r)}\geq \frac{M+m}{2},\qquad \mbox{ for }  r\in [\theta\, R, R].
\end{equation}
Since $R$ can be chosen arbitrarily large, we can suppose
$\theta R>R_0$ as well. Consider, for any $0<\varepsilon<(M-m)/4$, the lower barrier $w(r)=g(r)\,\widetilde\Gamma(r)$ where $g$ is the following step function
\[
g(r)=\begin{cases}
0&\text{if $r<R_0$},\\
m-\varepsilon_0&\text{if $R_0\leq r<\theta R$},\\
\frac{M+m}{2}&\text{if $\theta R\leq r<\sqrt{\theta}R$},\\
m+\varepsilon &\text{if $\sqrt{\theta}\,R<r$}.
\end{cases}
\]
It is easily seen that\footnote{As a set $E$ occuring in the definition $\widetilde{D}^{s,p}(\overline{B_R}^{\,c})$ one can take for example $E=\overline B_{\sqrt{\theta}\,R}^c$.} $w\in \widetilde{D}^{s,p}(\overline{B_R}^{\,c})$. Moreover,
by using \eqref{uvgeq}, it is readily verified that $w\leq U$ in $\overline{B_R}$. We claim that, for sufficiently small $\varepsilon_0$ and $\varepsilon$ and sufficiently large $R$,  it holds
\[
(-\Delta_p)^s w\leq (-\Delta_p)^s U,\qquad \mbox{ in }\overline{B_R}^{\,c}.
\]
This would end the proof, since Theorem \ref{comparison} would yield $U\geq w$ in $\mathbb{R}^N$ and then
\[
m=\liminf_{r\to +\infty} r^{\frac{N-s\,p}{p-1}}\,U(r)=\liminf_{r\to +\infty} \frac{U(r)}{\widetilde\Gamma(r)}\geq \liminf_{r\to +\infty} g(r)= m+\varepsilon,
\]
giving a contradiction.
The function $w-(m+\varepsilon)\,\widetilde \Gamma$ is supported in
$B_{\sqrt{\theta}\,R}\Subset B_{R}$ and thus using Proposition \ref{nonlocalb} with
\[
\Omega=\overline{B_R}^{\,c},\qquad u=(m+\varepsilon)\,\widetilde\Gamma,\qquad f= (-\Delta_p)^s\Big((m+\varepsilon)\,\widetilde\Gamma\Big) ,\qquad v=w-(m+\varepsilon)\,\widetilde\Gamma,
\]
and \eqref{splv}, for any $|x|>R$ it holds
\begin{equation}
\label{uno}
\begin{split}
(-\Delta_p)^s w(x)&= (m+\varepsilon)^{p-1}\,(-\Delta_p)^s\widetilde\Gamma(x)\\
&\quad+\int_{B_{\sqrt{\theta}\,R}}\frac{J_p\big((m+\varepsilon)\,\widetilde\Gamma(x)-w(y)\big)-J_p\big((m+\varepsilon)\,(\widetilde\Gamma(x)-\widetilde\Gamma(y))\big)}{|x-y|^{N+s\,p}}\, dy\\
&\leq \frac{C}{|x|^{N+s\,p}}+\int_{B_{\sqrt{\theta}\,R}}\frac{h(x, y)}{|x-y|^{N+s\,p}}\, dy,
\end{split}
\end{equation}
where
\[
h(x, y)=J_p\big((m+\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))\big)-J_p\big(w(y)-(m+\varepsilon)\,\widetilde\Gamma(x)\big).
\]
We now decompose the last integral in \eqref{uno} as follows
\begin{equation}
\label{scassamento}
\int_{B_{\sqrt{\theta}\,R}}\, dy=\int_{B_{R_0}}\, dy+\int_{B_{\theta\,R}\setminus B_{R_0}}\, dy+\int_{B_{\sqrt{\theta}\,R}\setminus B_{\theta\,R}}\, dy,
\end{equation}
and proceed to estimate each term separately.
\par
Being $R_0=R_0(\varepsilon_0)$ and $h$ universally bounded, it holds
\begin{equation}
\label{due}
\int_{B_{R_0}}\frac{h(x, y)}{|x-y|^{N+s\,p}}\, dy\leq \|h\|_{L^\infty(\mathbb{R}^{2\,N})}\,\frac{\omega_N\,R_0^N}{\Big||x|-R_0\Big|^{N+s\,p}}\leq \frac{C(\varepsilon_0)}{|x|^{N+s\,p}}\,\left(1-\theta\right)^{-N-s\,p},
\end{equation}
where we used that (recall that we are assuming $\theta\,R> R_0$)
\[
\Big||x|-R_0\Big|\ge \left(1-\frac{R_0}{R}\right)\,|x|\ge (1-\theta)\,|x|,\qquad \mbox{ for }|x|>R.
\]
For the second integral in \eqref{scassamento}, we notice that for $y\in B_{\theta\, R}\setminus B_{ R_0}$ and $x\in B_R^c$ we have
\[
h(x, y)=J_p\big((m+\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))\big)-J_p\big((m+\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))-(\varepsilon+\varepsilon_0)\,\widetilde\Gamma(y)\big).
\]
Observe that by \eqref{licciz}, with simple manipulations we get
\[
\begin{split}
h(x, y)&\le c\, \Big[(m+\varepsilon)^{p-2}+(\varepsilon+\varepsilon_0)^{p-2}\Big]\,(\varepsilon+\varepsilon_0)\,\widetilde \Gamma(y),
\end{split}
\]
for $x\in B_R^c$, $y\in B_{\theta R}\setminus B_{R_0}$ and $c=c(p)>0$. Therefore, since
\[
|x-y|\ge \Big||x|-|y|\Big|\ge |x|-\theta\,R\ge \left(1-\theta\right)\,|x|,\qquad \mbox{ for } x\in  B_R^c,\ y\in B_{\theta\, R},
\]
recalling the definition of $\widetilde\Gamma$ we get
\begin{equation}
\label{tre}
\begin{split}
\int_{B_{\theta R}\setminus B_{R_0}}\frac{h(x, y)}{|x-y|^{N+s\,p}}\, dy&\leq \frac{C\,(\varepsilon+\varepsilon_0)}{(1-\theta)^{N+s\,p}\,|x|^{N+s\,p}}\int_{B_{\theta R}\setminus B_{ R_0}}\frac{1}{|y|^{N-s\,p}}\, dy\\
&\leq  C\,\frac{(\varepsilon+\varepsilon_0)}{(1-\theta)^{N+s\,p}}\,\frac{(\theta\,R)^{s\,p}}{|x|^{N+s\,p}},
\end{split}
\end{equation}
where $C=C(N,s,p,M+m)>0$.
For the third integral in \eqref{scassamento}, for $y\in B_{\sqrt{\theta}R}\setminus B_{\theta R}$ we have
\[
\begin{split}
h(x, y)&=J_p\Big((m+\varepsilon)(\widetilde\Gamma(y)-\widetilde\Gamma(x))\Big)-J_p\left((m+\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))+\left(\frac{M-m}{2}-\varepsilon\right)\,\widetilde\Gamma(y)\right)\\
&\leq J_p\Big(\underbrace{(m+\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))}_a\Big)-J_p\Big(\underbrace{(m+\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))}_{a}+\underbrace{\left(\frac{M-m}{4}\right)\,\widetilde\Gamma(y)}_b\Big),
\end{split}
\]
since $\varepsilon<(M-m)/4$. The inequality \eqref{27} thus gives
\[
h(x, y) \leq -2^{2-p}\,\left(\frac{M-m}{4}\right)^{p-1}\,\widetilde\Gamma(y)^{p-1}.
\]
Therefore,  using
\[
|x-y|\leq 2\,|x|,\qquad \mbox{ for } |x|>R, \ |y|<\sqrt{\theta}\,R,
\]
we obtain
\begin{equation}
\label{quattro}
\begin{split}
\int_{B_{\sqrt{\theta}\,R}\setminus B_{\theta\, R}}\frac{h(x, y)}{|x-y|^{N+s\,p}}\, dy &\leq -\frac{c\,(M-m)^{p-1}}{|x|^{N+s\,p}}\int_{B_{\sqrt{\theta}\,R}\setminus B_{\theta\, R}} |y|^{s\,p-N}\, dy\\
&\leq -c\,\theta^\frac{s\,p}{2}\,\Big(1-\theta^\frac{s\,p}{2}\Big)\,(M-m)^{p-1}\,\frac{R^{s\,p}}{|x|^{N+s\,p}},
\end{split}
\end{equation}
for a constant $c=c(N,s,p)>0$.
Gathering toghether the estimates \eqref{uno}, \eqref{due}, \eqref{tre} and \eqref{quattro} we proved
\[
\begin{split}
(-\Delta_p)^s w(x)&\leq \left(C+\frac{C(\varepsilon_0)}{(1-\theta)^{N+s\,p}}\right)\,\frac{1}{|x|^{N+s\,p}}\\
&\quad -\left[c\,\Big(1-\theta^\frac{s\,p}{2}\Big)\,(M-m)^{p-1}-\frac{C\,(\varepsilon+\varepsilon_0)}{(1-\theta)^{N+s\,p}}\right]\,\frac{R^{s\,p}\,\theta^{s\,p}\,}{|x|^{N+s\,p}}.
\end{split}
\]
Thus we can choose $\varepsilon+\varepsilon_0$ small enough (depending only on $N,p,s,M-m$ and the chosen minimizer $U$), so that the second term above is negative.
For any such a choice we have, for any $|x|> R$,
\[
(-\Delta_p)^s w(x)\leq \frac{C(\varepsilon_0)}{|x|^{N+s\,p}},\qquad (-\Delta_p)^s U(x)= U(x)^{p^*-1}\geq \frac{1}{C\,|x|^{N+\frac{s\,p}{p-1}}},
\]
where in the last estimate we used Corollary \ref{lower-Dec}.
Since $p> 2$, for sufficiently large $R$ it holds
\[
 \frac{1}{C\,|x|^{N+\frac{s\,p}{p-1}}}\geq \frac{R^{s\,p\,\frac{p-2}{p-1}}}{C\,|x|^{N+s\,p}}\geq \frac{C\,(\varepsilon_0)}{|x|^{N+s\,p}},
\]
and thus the claim follows.
\vskip.2cm\noindent
$\bullet$  \underline{\em Case $1<p<2$}. There exists $R_0=R_0(\varepsilon_0)>2$ such that
\[
\frac{U(r)}{\widetilde\Gamma(r)}\leq M+\varepsilon_0,\quad \mbox{ for } r\geq R_0
\]
and we can choose an arbitrarily large $R>R_0$ such that
$$
\frac{U(R)}{\widetilde\Gamma(R)}\leq m+\frac{M-m}{4}.
$$
As before, we consider $\delta=(M-m)/4$ in Lemma \ref{lm:intervalli}: there exists $\theta<1$ so that for any such $R$ it holds
\begin{equation}
\label{uvleq}
\frac{U(r)}{\widetilde\Gamma(r)}\leq \frac{M+m}{2},\qquad \mbox{ for every } r\in [R, R/\theta].
\end{equation}
Since $U\in L^\infty(\mathbb{R}^N)$, there exists $\overline C>0$ such that $U\leq \overline C\, \widetilde\Gamma$ in $\mathbb{R}^N$, then for any $0<\varepsilon<(M-m)/4$ we consider the upper barrier $w(r)=g(r)\,\widetilde\Gamma(r)$, where
\[
g(r)=
\begin{cases}
\overline C&\text{if $r<R_0$},\\
M+\varepsilon_0&\text{if $R_0\leq r<R$},\\
\frac{M+m}{2}&\text{if $R\leq r<R/\sqrt{\theta}$},\\
M-\varepsilon &\text{if $R/\sqrt{\theta}<r$}.
\end{cases}
\]
Again, it is easy to verify that $w\in \widetilde{D}^{s,p}(\overline{B_R}^{\,c})$.
Using \eqref{uvleq}, we can verify that $w\geq U$ in $B_{R/\theta}$. We claim that, for sufficiently small $\varepsilon_0$ and $\varepsilon$ and sufficiently large $R$,  it holds
\[
(-\Delta_p)^s w\geq (-\Delta_p)^s U,\qquad \mbox{ in } B^c_{R/\theta}.
\]
This would end the proof, since the comparison principle of Theorem \ref{comparison} would yield $U\leq w$ in $\mathbb{R}^N$ and then
\[
M=\limsup_{r\to +\infty} r^{\frac{N-s\,p}{p-1}}\,U(r)=\limsup_{r\to+\infty} \frac{U(r)}{\widetilde\Gamma(r)}\leq \limsup_{r\to +\infty} g(r)= M-\varepsilon,
\]
which gives again a contradiction.
The function $w-(M-\varepsilon)\,\widetilde\Gamma$ is supported in
$B_{R/\sqrt{\theta}}\Subset B_{R/\theta}$ and thus using again Proposition \ref{nonlocalb}
with
\[
\Omega=\overline{B_{R/\theta}}^{\,c},\qquad u=(M-\varepsilon)\,\widetilde \Gamma,\qquad f= (-\Delta_p)^s\Big((M-\varepsilon)\,\widetilde\Gamma\Big) ,\qquad v=w-(M-\varepsilon)\,\widetilde \Gamma,
\]
and \eqref{splv}, for any $|x|>R/\theta$ it holds
\begin{equation}
\begin{split}
\label{unobis}
(-\Delta_p)^s w(x)&= (M-\varepsilon)^{p-1}\,(-\Delta_p)^s\,\widetilde\Gamma(x) \\
&\quad+\int_{B_{R/\sqrt{\theta}}}\frac{J_p\big((M-\varepsilon)\,\widetilde\Gamma(x)-w(y)\big)-J_p\big((M-\varepsilon)\,(\widetilde\Gamma(x)-\widetilde\Gamma(y))\big)}{|x-y|^{N+s\,p}}\, dy \\
&\geq \frac{1}{C\,|x|^{N+sp}}+\int_{B_{R/\sqrt{\theta}}}\frac{h(x, y)}{|x-y|^{N+sp}}\, dy,
\end{split}
\end{equation}
where
\[
h(x, y)=J_p\big((M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))\big)-J_p\big(w(y)-(M-\varepsilon)\,\widetilde\Gamma(x)\big).
\]
As above, we now decompose the last integral in \eqref{unobis} as
\[
\int_{B_{R/\sqrt{\theta}}}\, dy=\int_{B_{R_0}}\, dy+\int_{B_{R}\setminus B_{R_0}}\, dy+\int_{B_{R/\sqrt{\theta}}\setminus B_{R}}\, dy,
\]
and proceed to estimate each term separately.
\par
Being $R_0=R_0(\varepsilon_0)$ and $h$ universally bounded, as before we get
\begin{equation}
\label{duebis}
\int_{B_{R_0}}\frac{h(x, y)}{|x-y|^{N+sp}}\, dy\geq -\frac{C(\varepsilon_0)}{|x|^{N+s\,p}},
\end{equation}
where this time we used that (recall that we are assuming $R>R_0$)
\[
\Big||x|-R_0\Big|\ge \left(1-\frac{R_0}{R}\,\theta\right)\,|x|\ge (1-\theta)\,|x|,\qquad \mbox{ for }|x|>R/\theta.
\]
For $y\in B_{R}\setminus B_{ R_0}$ we have
\[
h(x, y)=J_p\Big((M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))\Big)-J_p\Big((M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))+(\varepsilon+\varepsilon_0)\,\widetilde\Gamma(y)\Big),
\]
and by subaddivity of $\tau\mapsto \tau^{p-1}$, we get
\[
h(x, y)\geq -(\varepsilon+\varepsilon_0)^{p-1}\,\widetilde\Gamma(y)^{p-1}.
\]
Therefore, the analogue of \eqref{tre} is now
\begin{equation}
\label{trebis}
\int_{B_{R}\setminus B_{ R_0}}\frac{h(x, y)}{|x-y|^{N+sp}}\, dy\geq-C\,(\varepsilon+\varepsilon_0)^{p-1}\frac{R^{s\,p}}{|x|^{N+s\,p}},
\end{equation}
and again $C=C(N,s,p,M+m)>0$. For the previous estimate we also used that
\[
|x-y|\ge |x|-R\ge \left(1-\theta\right)\,|x|,\qquad \mbox{ for } |x|>R/\theta,\ |y|<R.
\]
For $y\in B_{R/\sqrt{\theta}}\setminus B_{R }$ and $x\in B^c_{R/\theta}$ we have
\[
\begin{split}
h(x, y)&=J_p\Big(\underbrace{(M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))}_a\Big)\\
&-J_p\Big(\underbrace{(M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))}_a-\underbrace{\left(\frac{M-m}{2}-\varepsilon\right)\,\widetilde\Gamma(y)}_b\Big).
\end{split}
\]
Clearly
\[
0\leq a=(M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))\leq (M-\varepsilon)\,\widetilde \Gamma(y)=:A,
\]
so that \eqref{ineqb} provides
\[
h(x, y)\geq \max\left\{(M-\varepsilon)^{p-1}-\left(\frac{M+m}{2}\right)^{p-1},\,\left(\frac{M-m}{2}-\varepsilon\right)^{p-1}2^{1-p}\right\} \,\widetilde\Gamma(y)^{p-1}.
\]
Proceeding as for \eqref{quattro} and using
\[
|x-y|\leq 2\,|x|,\qquad \mbox{ for } x\in B^c_{R/\theta}, \ y\in B_{R/\sqrt{\theta}},
\]
we thus obtain
\begin{equation}
\label{quattrobis}
\int_{B_{R/\sqrt{\theta}}\setminus B_{R}}\frac{h(x, y)}{|x-y|^{N+s\,p}}\, dy \ge \frac{c}{|x|^{N+s\,p}}\int_{B_{R/\sqrt{\theta}}\setminus B_{R}} |y|^{s\,p-N}\, dy \ge c\,\frac{R^{s\,p}}{|x|^{N+s\,p}},
\end{equation}
for a small constant $c$ depending only on $M$ and $m$.
Gathering together the estimates \eqref{unobis}, \eqref{duebis}, \eqref{trebis} and \eqref{quattrobis}, we proved
\[
(-\Delta_p)^s w(x)\geq -\frac{C(\varepsilon_0)}{|x|^{N+s\,p}}+\Big(c-C\,(\varepsilon+\varepsilon_0)^{p-1}\Big)\,\frac{R^{s\,p}}{|x|^{N+s\,p}}.
\]
in $B_{R/\theta}^c$. We can thus choose $\varepsilon_0$ and $\varepsilon$ small enough so that the second term above is positive. For any such choice we have, for any $|x|>  R/\theta$,
\[
(-\Delta_p)^s w(x)\geq -\frac{C(\varepsilon_0)}{|x|^{N+s\,p}}+\frac{c}{2}\frac{R^{s\,p}}{|x|^{N+s\,p}},
\]
and for sufficiently large $R$ so that $c\,R^{s\,p}>4\,C(\varepsilon_0)$ it holds
\[
(-\Delta_p)^s w(x)\geq \frac{c}{4}\frac{R^{s\,p}}{|x|^{N+s\,p}}.
\]
By using Corollary \ref{lower-Dec} and the fact that $1<p<2$, for every $|x|\ge R/\theta$ we get
\[
(-\Delta_p)^s U(x)= U^{p^*-1}(x)\leq \frac{C}{|x|^{N+\frac{s\,p}{p-1}}}\leq \frac{C\,\theta^{s\,p\,\frac{2-p}{p-1}}}{R^{s\,p\,\frac{2-p}{p-1}}|x|^{N+s\,p}}.
\]
We thus conclude that $(-\Delta_p)^sU\leq (-\Delta_p)^s w$ in $\overline{B_{R/\theta}}^{\,c}$ for $R$ sufficiently large, as desired.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Odorrana hejiangensis est une espèce d'amphibiens de la famille des Ranidae.

Répartition
Cette espèce est endémique du xian de Hejiang dans la province du Sichuan en République populaire de Chine.

Étymologie
Son nom d'espèce, composé de hejiang et du suffixe latin , « qui vit dans, qui habite », lui a été donné en référence au lieu de sa découverte, Shunyangxi dans le xian de Hejiang.

Publication originale
 Deng & Yu, 1992 : A new species of the genus Rana from China. Journal of Sichuan Teachers College (Natural Science), , .

Liens externes

Notes et références

Anoure (nom scientifique)
Ranidae
Faune endémique du Sichuan

redpajama_set_name: RedPajamaWikipedia


text:
Scouting is active on Guadeloupe and on Saint Martin, connected administratively in the Territoire de Guadeloupe et Saint Martin of the Scouts et Guides de France. The Scouts of the islands participate in many Caribbean regional Scout camps, jamborees and other Scout activities. Scouting closely follows the program of the Scouts et Guides de France. The Scouts et Guides de Saint Martin and the Scouts et Guides de Guadeloupe are subsections of the larger regional entity.

The Scout emblem incorporates elements of the old coat of arms of Guadeloupe, as well as the old arms of French Guiana and Martinique.

See also
 Scouting Antiano (Dutch Scouting on the divided island of Saint Martin)

Guadeloupe
History of the Collectivity of Saint Martin
History of Guadeloupe
Guadeloupe, Saint Martin
Guadeloupe


Attractive 4 Bdrm home located along the 18th fairway w/ view's to the East.Exceptional out door area w/ Lg covered patio, stone fireplace, rock & wrought iron fence. Spacious interior w/ 2 Lg living area's, 2 fireplaces, formal dining Rm., breakfast rm., open kitchen, 3 main level Bdrms & upper Bunk/game rm w/ wet bar & bath. Side-entry 3 car garage.


www.skinspa.co.uk is a site operated by Skin Spa LTD ("We"). We are registered in England and Wales under company number 08035827 and have our registered office at SO31 4JW. Our main trading address is 13 Bell Street, Romsey, SO51 8GY.
If you know or suspect that anyone other than you knows your user identification code or password, you must promptly notify us at info@skinspa.co.uk.


text:
Q: How do I create a directory in all subdirectories? Suppose I have a directory structure like this:

$ [~/practice] ls
a/ b/ c/ d/

Now I want to create a directory tmp1 in all sub directories of practice and I do this:

$ [~/practice] mkdir */tmp1
mkdir: cannot create directory `*/tmp1': No such file or directory

Then I try the -p switch and I endup with a directory named * with a sub directory tmp1

$ [~/practice] mkdir -p */tmp1

$ [~/practice] ls
*/ a/ b/ c/ d/

I know the use of -p switch is to create multiple nonexistent directories. I just thought it might help.
How do I create tmp1 in all subdirectories at once?
If this can be done, how do I extend it to create \tmp1, \tmp2, \tmp3 in \a, \b and \c at once?
Edit: I missed mentioning that the directories don't have to be simple and in order, like a, b, c etc., and the directory to be created is not necessarily like tmp1, tmp2.

$ [~/practice] ls
dog/ cat/ rat/

In them, I would like to have something like

$ [~/practice] ls *
dog:
red/ blue/

cat:
red/ blue/

rat:
red/ blue/

A: Off topic since you're mentioning bash, but for the record, with zsh, you'd do:
dirs=(*(/))
mkdir -- $^dirs/tmp1

$^var turns on brace-like expansion for the expansion of the array. It's reminiscent of rc's ^ operator and in rc (or its derivative es), you'd write it:
dirs=(*/)
mkdir -- $dirs^tmp1

However note (and the same applies to the bash solutions given here) that in the rc solution dirs would also contain symbolic links to directories. In the zsh solution, change *(/) to *(-/) if you want to include symlinks to directories.

A: With globs :
for dir in */; do mkdir -- "$dir/tmp1"; done

NOTE

*

*I treat only dirs (including symlinks to dirs) with the little hack of using */ as a glob

*If you want to create multiple subdirs at once :
for dir in */; do mkdir -- "$dir"/{tmp1,foo,bar,qux}; done

A: Portably, loop over the parent directories:
for d in */; do mkdir "$d/red" "$d/blue"; done

Add -- after mkdir if you may have directories whose name starts with -.
In zsh, you can do it in a single command with the e glob qualifier:
mkdir *(/e\''REPLY=($REPLY/{red,blue})'\')

but it's quicker to type this as two commands:
d=(*(/)); mkdir $^d/{red,blue}

A:
[...] how do I extend it to create \tmp1, \tmp2, \tmp3 in \a, \b and \c at once?

 mkdir {a,b,c}/tmp{1,2,3}

A: With GNU or BSD* find:
find -mindepth 1 -maxdepth 1 -type d -exec mkdir {}/newdir \;

or using parameter expansion:
dirs=(*/)
mkdir -- "${dirs[@]/%/newdir}"

*Includes OS X

A: Variation on sputnick's answer which avoids non-directory files:
for x in *; do if [ -d "$x" ]; then mkdir "$x/tmp1"; fi; done

A: You can do that like that:
mkdir {a,b,c,d}/tmp1

A: This is a good one from the old Unix Masters (tested on FreeBSD 10.1) if you want to version control (with RCS check in locked: ci -l with a brief comment as ir: for initial release) your /etc/* config files on ~/.his/etc
sudo rsync -av /etc/ ~/.his
sudo find -d ~/.his/etc -type d -exec echo '{}/RCS' \; | xargs sudo mkdir -p
sudo find -d ~/.his/etc \! -name "RCS" | xargs sudo ci -l -mir -t-ir

A: Should be \! name "RCS" and it works best on the first check in.
sudo find ~/.his/etc \! -name "RCS" -ls
sudo find -d ~/.his/etc \! -name "RCS" | xargs sudo ci -l -mir -t-ir
This last one also works very well for the cjeck in part filtering all RCS content on the file name:
sudo find -d ~/.his/etc -type f | grep -v RCS | xargs sudo ci -l -mir -t-ir

A: this is a great help to me. i had a need to create a set of subdirectories in multiple folders and something like:
!/bin/sh

cd /dest/cont

for dir in */*/; do
    mkdir -p -- $dir/{FB,Video,Audio,proj};
done

worked great except when a white space is encountered. is there a way to ignore spaces in the */*/ part of the arguments?

redpajama_set_name: RedPajamaStackExchange


Über news. Source needs the same vitamin injection by SourceMod as amxmodx did for all HL1 mods. But I bet we wont see it in Steam's news page. I dont know where you get the energy and inspiration from to keep doing what you do, but its awsome. GJ!
sweet news. if the new pawn will be close to half life 1 pawn, i will be happy to give source mod a try again.
does that mean that u won't rehook events when a cvar is changed for example or like unhook events ?
It means that you can optimize your plugin by removing hooks when they are no longer needed.


Thankful for all the honest people in my life many whom I've met in life groups at my home church.
Love your heart filled writing … write on girl!!!
Thank you Marissa! I'm so grateful for your encouragement – you are right that people are the greatest gift we're ever given – we need them desperately! Merry Christmas and thank you for your ministry through DIB.


text:
Norwalk-La Mirada USD selects Pearson's PowerSchool® Premier as new student information system
Pearson announced that the Norwalk-La Mirada USD has selected Pearson's PowerSchool as its new web-based student information system (SIS)
Rancho Cordova, CA (PRWEB) October 20, 2009 -- Pearson announced today that the Norwalk-La Mirada Unified School District (USD) has selected Pearson's PowerSchool as its new web-based student information system (SIS). Pearson's PowerSchool will serve Norwalk-La Mirada USD's more than 21,000 students.
Norwalk-La Mirada USD selected Pearson's PowerSchool as its comprehensive SIS based on several factors, including flexibility and customizability. Having narrowed the field to two finalists, Norwalk-La Mirada USD chose Pearson after PowerSchool delivered a user-friendly showing during the sandbox experience, where district personnel were able to use the system in a demonstration environment.
District Superintendent, Dr. Ruth Pèrez, summed up the selection process by stating, "The district spent a year analyzing district needs and creating a Request for Proposal (RFP). We received nine responses to the RFP from reputable companies across the nation. After careful analysis of all RFPs, software previews, and investigations by a group of 26 people representing all stakeholders, PowerSchool was selected as the software that best fit our needs because of ease-of-use and ability to share reports with other districts in the PowerSchool community. PowerSchool is very complete as it stands but also allows user modifications that are not disturbed by vendor upgrades. Pearson's responses to questions in the RFP seem to show that the company has a solid understanding of California state reporting requirements. Pearson is recognized worldwide as an education company that is financially stable."
PowerSchool will serve as the information backbone for Norwalk-La Mirada USD. Through its robust set of communications tools, PowerSchool will provide access to secure, real-time student information through a web browser, effectively creating an interactive dialogue between parents, students, and teachers. Additional benefits, such as master scheduling and reporting capabilities, will lend substantial efficiencies to the district.
"We are very pleased to welcome Norwalk-La Mirada as a PowerSchool customer," said Paul Fletcher, President of the School Systems group of Pearson. "More districts are using PowerSchool than any other student information system available on the market. Norwalk-La Mirada is joining thousands of other districts in choosing PowerSchool, which enables districts to adapt to the ever-evolving education technology landscape."
Norwalk-La Mirada USD expects to complete the implementation of PowerSchool by January 2010. For more information about Pearson and PowerSchool, please visit www.PearsonSchoolSystems.com.
Pearson, the global leader in education and education technology, reaches and engages today's digital natives with effective and personalized learning, as well as dedicated professional development for their teachers. This commitment is demonstrated in the company's investment in innovative print and digital education materials for preK through college, student information systems and learning management systems, teacher professional development, career certification programs and testing and assessment products that set the standard for the industry. The company's respected brands include Scott Foresman, Prentice Hall, Addison Wesley, Benjamin Cummings, PEMSolutions, Stanford 10, SuccessNet, MyLabs, PowerSchool, SuccessMaker, and many others. Pearson's comprehensive offerings help inform targeted instruction and intervention so that success is within reach of every student at every level of education. Pearson's commitment to education for all is supported by the global philanthropic initiatives of the Pearson Foundation. Pearson's other primary businesses include the Financial Times Group and the Penguin Group.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Importing a whole folder of python files In the current python program I'm working on, I need to access a lot of stored data. I store it in the form of a bunch of dictionaries, each in their own file. Each file has a single command: giveArchive(). So to access one of the files, I use:
import fileName
return fileName.giveArchive()

And this has worked well so far, but as the number of files I need grows, I want to streamline this a little bit. I'd like to store all of these files in the same folder, and that folder in the same directory as my main file. Is there some way I can import every file in a folder? And if I do, how can I use 'giveArchive()' from specific files in it?

A: You can do something like:
from folder.subfolder.deepersubfolder import filename
return filename.giveArchive()

this assumes folder can be accessed from the directory your script is running in


It looks like great progress is being made on spur reduction, so back to LPF switching and a few other things. Received the first set of boards and have already made some modifications for better wireability.
1) I decided to eliminate R3 off the uBITX board and instead add it to the LPF Relay Board because it frees up space on the uBITX and allows more efficient RF wiring for receive. Since the "old R3" also switches M1 to M2, I'm changing that to how it's wired on the V4 uBITX by adding a 2N7000, changing R70 to 1K, add a 1K R78, add a 1uF C79, add a D14, and driving the 2N7000 with a line over to T/R (a 2N7000 driven directly by T/R is faster than any speeding relay contact).
2) Replace 3 sets of dual 2N3904 Transistors in the PA driver with 3 sets of dual 2N2222A Transistors. Allison had suggested changing the emitter resistors from 22 ohms to 10 ohms (for dual 2N2222A's ?
3) Add a three 1N4148 diode-OR from TxA,TxB, and TxC driver inputs to a TxD driver to pick that relay, which removes the 10m/12m/(15m) LPF when inserting the proper (A/B/C) LPF during Tx. Yes, that means 4 relays are active during TxA,TxB, or TxC transmit. All four of the LPFs use the existing inductor/capacitor components off the uBITX board. They are just moved over to a small 1.5" x 0.5" bare LPF board (that footprint is also the one QRP labs has for their LPF/BPF filters, but they don't sell blank boards.
4) uBITX Antenna will be wired to either one of two SMA connector pads.
5) T11 will be connected to the RF input on the LPF Relay Board with about 1" of coax.
6) Found some of the smaller 12V 10 pin DPDT bifurcated contact relays on ebay for about $1 each. They fit nicely.
7) Leave all the traces, just make a cut and remove 1/16" of the trace where needed.
8) Will provide pictures later.


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Since this is the very first time that I've ever auctioned anything, my auction page [Link] needed some editing.
➜ Auction ends at 12 noon - Wednesday, November 14th.
➜ All items are sold as is, where is.
Tomorrow afternoon, an RV dealer has an appointment with me to take a look at MsTwayler and make an offer to buy her.
I am in NO hurry to sell MsTwayler. I have tons of things to do while at Lone Wolf, and am having a great time being here.
Everything shall happen at the right time 😎.
Just after towing Scampy from her parking spot at Lone Wolf over to where MsTwayler's site is located, our Samlex inverter's [Link] low voltage alarm began sounding!
Going outside to check the battery connections from the Renogy Rover solar charger [Link] to Scampy's House Battery, everything looked OK.
However, when we checked the battery voltage readout of the Rover, it showed that the battery had only an 11.7 voltage reading! A very low voltage reading!
A voltage reading is not an accurate indicator of a battery's state of charge. So, we did a specific gravity test which showed that our House Battery had a 1.17 specific gravity reading. Bad, bad, bad!
That was when it was noticed that Transito's roof mounted solar panel was in the shade of the Sun! For goodness sakes! Of all the things?
But even so, we only parked in the shade a few minutes ago. For the battery to get so bad, so fast, means that the battery very likely needs replacement.
What's the best for a House Battery?
For many years, the Trojan T-105 6-volt deep cycle battery has been at the top of the heap!
However, the Scampy Team does not need a lot of electric power. What the Scampy Team needs is a small amount of dependable electric power.
Emphasis on the word: "Dependable"
Mighty Max ML35-12 - 12V 35AH U1 Deep Cycle AGM Solar Battery.
Would you please take a look at it? Click [Here] to view.
I sure would love to receive some feed-back about the KoolerAir.
At the time that I Blogged about KoolerAire, I did not understand that the ice chest in which KoolerAire is installed, must be filled with a couple of bags of ice cubes.
That ice is where the coolness comes from. I found a YouTube video which demonstrates the KoolerAire device.
The guy in that video said that he bought two bags of ice.
14 lbs which cost $5.38!
Click [Here] to view that video.
That guys review was quite funny. Especially when his van increased by 7 degrees and he measured the cooler air going about 8 inches away from the thing.
Air conditioner's are often rated in tons of cooling. One ton of ice produces the equivalent of 12,000 BTUs over a 24 hour period, or 24,000 BTUs for 12 hours, etc.
A typical RV air conditioner produces about 12,000 BTUs.
How much cooling do you think you'd get from a 5 pound bag of ice? Not much.
Your comment states exactly how I feel.
Good luck with selling your stuff, that is a good price for your Weber Q 1000, we love ours best BBQ ever.
An AGM battery may be best for you because of the maintenance issues of flooded batteries but 35AH doesn't sound like nearly enough AH even for just LED lighting.
Consider getting more battery than you need. A larger battery won't cycle as deeply in normal use, and the shallower you cycle a battery, the longer it will last.
Plus you'll have reserve power to carry you through an unexpected cold/dark period, etc.
A pair of 6 volt golf cart batteries are hard to beat on a dollar per watt basis and they're very durable as long as you keep an eye on their water so it doesn't fall below the level of the plates.


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Homes | Property
See The World's First-Ever Sky Pool At A London Luxury Residence
By Kaitlyn McInnis
Cover The Sky Pool is part of new luxury London apartment complex, Embassy Gardens (Photo credit: Embassy Gardens)
The 82-foot acrylic pool, part of luxury property development Embassy Gardens, is suspended nearly 115 feet above London's rapidly gentrifying Nine Elms neighbourhood
The world's first-ever sky pool has this week opened its waters to architecture and swim enthusiasts across London. The impressive aquatic structure, known as Sky Pool, is suspended about 115 feet in the air, offering a wildly daring and architecturally striking alternative to the typical city swimming pool.
Located between the two apartment towers at Embassy Gardens, the floating acrylic Sky Pool comes as a unique alternative to the traditional rooftop pool. The aquatic offering is not for the faint of heart though—it runs about 82 feet in length with nearly 46 feet completely suspended in mid-air.
According to a recent Robb Report story, the stunning project comes as a collaboration between architect firm Arup, engineers Eckersley O'Callaghan, and acrylic fabricator Reynolds Polymer Technologies. Arup's lengthy resume includes well-known projects around the globe, including the Jewel Changi Airport in Singapore and the TWA Flight Center in New York.
See also: The Coolest Hotel Pools In Malaysia
Being that the aquatic project is connected to two prestigious apartment towers in the city, the avant-garde pool structure is not currently opened to the public. Those interested in taking a dip will need a membership. In fact, all residents of the building will be enrolled in the Embassy Gardens' wellness programme, which includes access to the water feature, a cutting edge gym, movie theatre, spa and lounge, rooftop terrace, and indoor pool.
Interested in finding out more about the condo project or wellness membership? More information about the wellness membership, including pricing, is available on the official Embassy Gardens' website. For those interested in buying a unit in the prestigious tower project, prices start at about US$1.4 million (RM5.8 million).
See also: Introducing Tatler House: An Exclusive Venue At The Mandarin Oriental, Kuala Lumpur
Luxury property london property london pools luxury pools


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Bison Media Zone
Government and Politics Jan 17th 2020 - 7am
Vanity plate requests taken very SRIUSLY in North Dakota
Letters Jan 18th 2020 - 10am
Shaw: Remembering the Jim Crow South
ND AG issues cease-and-desist on widespread 'Irish Traveller' construction scams
Weather Jan 17th 2020 - 4pm
See blizzard photos and videos of snow drifts and road closures
Family's medical riddle solved with genetic testing
FARGO - Karen Daggett was enjoying a five-course Valentine's Day dinner with friends when she suddenly felt her heart beating erratically and her vision clouded.
Written By: Patrick Springer | Aug 17th 2015 - 7am.
Susan Bertelson, left, Karen Nelson Daggett, and Janee Nelson, photographed at Island Park in Fargo, N.D., on Monday, August 10, 2015. Nick Wagner / The Forum
FARGO – Karen Daggett was enjoying a five-course Valentine's Day dinner with friends when she suddenly felt her heart beating erratically and her vision clouded.
"Something is really wrong," she told her husband during the salad course. "We need to go to a hospital."
Fog shrouded the road, but the Frazee, Minn., couple found the hospital in Naples, Fla., where they were vacationing. Once in the emergency room, she was unable to provide her birthdate.
Despite the obvious urgency of her medical plight, Daggett had no inkling that the unnerving episode would reveal a hidden health complication that runs rampant in her family.
After being given intravenous fluids, and told to drink something "as fast as you can," Daggett was able to recall her birthdate.
But the doctors were unable to diagnose the cause of her problem during a five-day hospital stay. When discharged, she was advised to get to the Mayo Clinic, and soon - no later than 48 hours.
There, following a barrage of tests, a process of elimination pinpointed her problem: Daggett lacked a gene allowing her liver to metabolize a long list of medications.
It turned out she became dangerously dehydrated, apparently because her body went haywire after receiving medications for a heart procedure a couple of months earlier. She had critical levels of potassium, magnesium and calcium.
Drugs that aren't metabolized can accumulate in the body, sometimes reaching harmful levels.
"You become toxic and eventually all the organs start shutting down," Daggett said. "I'd be dead without Mayo."
Gene issue affects many
The revelation six years ago that Daggett's liver lacked an important protein enzyme to enable her to metabolize many medications since has rippled through several generations of her extended family, originally from Lake Park, Minn.
So far, 15 family members - including all five of her siblings - have been found to lack the important enzyme and two of her friends also have been found to have the condition, which is more common in those with Scandinavian and northern European ancestry.
"Those who have been tested have it," said Daggett's younger sister Janee Nelson, who lives in Lake Park. "There have been several who haven't been tested."
Having the diagnosis, made available through genetic testing, helped explain medical problems that for years plagued Daggett, her siblings and other family members.
Problems with irregular heartbeat are pervasive in the family, for example. So is the ineffectiveness of many painkillers, a complication that afflicted Daggett and her siblings.
Another sister, Susan Bertelson of Brookings, S.D., recalls experiencing a lot of pain during childbirth when she was younger.
"They were pumping me with stuff," she said, referring to pain medication. "Things just weren't working."
Once, Bertelson woke up during surgery. "I just said 'ouch,' " she said. "You wake up because it hurts."
Those who have the genetic mutation lack an important liver enzyme. They cannot metabolize acetaminophen, the active ingredient in Tylenol, for instance, and Daggett had adverse reactions to blood pressure medications until testing led her to a safe pill.
When the body doesn't properly absorb medications, they can accumulate in high levels and become toxic. Daggett's daughter once was given a high dose of Tylenol, and it almost killed her, she said.
In the case of Daggett and her sisters, it wasn't until they got older and confronted more health issues that the complications from the mutation became glaring.
By one estimate, 7 percent of whites and 2 percent to 7 percent of blacks are poor metabolizers of drugs that depend on an enzyme called CYP2D6 - the enzyme Daggett and her siblings lack altogether - a pathway that metabolizes antidepressants and opioids. That enzyme is involved in metabolizing up to a quarter of all drugs.
Similarly, one in five Asians is a poor metabolizer of drugs that depend on the enzyme CYP2C19, needed to metabolize the anti-seizure medication Dilantin, the sedative phenobarbital and Prilosec, used to treat frequent heartburn, among other drugs.
"Most people don't have a problem until they start a medication," Daggett said. "It's a very interesting adventure. I've almost died a couple of times."
'Knowledge is power'
A mental alarm started clanging when Daggett was awakened by the knowledge that she had inherited the genetic mutation. If she had it, other members of her family could be at risk.
Armed with information, she learned that the problem had been discovered only a few years before her diagnosis, by doctors at Mayo Clinic who learned that some women were unable to metabolize tamoxifen, an antiestrogen drug that is a mainstay in treatment of breast cancer.
Mayo researchers determined that those women would not be protected against the recurrence of the cancer and therefore should not take the drug - information that was especially of interest, since her sister Susan was diagnosed with breast cancer two weeks after Daggett learned she had the genetic variance.
Bertelson had DNA testing and learned that she, too, had the variance. As a result, her treatment did not include tamoxifen; she has not had a recurrence of her cancer.
"Knowledge is power," Daggett said.
Upon hearing that two of her sisters had the troublesome gene variance, Nelson immediately went in for testing, and learned that she also did.
Subsequent genetic testing has found that the variance has been found in four generations of the Nelson family - including both of the siblings' parents, now deceased.
Interestingly, three of the siblings' sets of great-grandparents came from the same village in Sweden, Appelbo, about 200 miles northwest of Stockholm. That might help to explain why the variance is so prevalent in the family, the sisters said.
Testing more common
At first, few of their health providers knew about how the missing enzyme could complicate their medical care. But in the years since, as the emerging field of pharmacogenetics has grown and become better known, physicians are more familiar with it.
Still, navigating a treatment course in a way that avoids adverse drug interactions, therapeutic failure or toxicity can be complicated, the sisters said.
"Our pharmacists are our best friend at this point," said Daggett, who once carried a list of all the medications she can't take to present to doctors unfamiliar with her condition.
Now, the sisters said, because the variance is becoming better known, health providers are better versed in how people lacking important enzymes might react to certain medications.
Clinical genetic testing is also becoming more common, including Sanford's Imaginetics program, Daggett said.
There are more than 50 of the CYP450 enzymes, but six enzymes metabolize 90 percent of drugs, according to a study. Some medical centers now routinely test heart patients for the enzyme, she said.
The Nelson sisters are telling their story in the hope that others who have had similar problems with medications might be prompted to seek testing to determine if they lack important enzymes.
"This discovery has covered four generations of my family and I am only sorry that we weren't able to help my parents in their end-of-life transitions, but we just didn't know," Daggett said.
Anyone suspecting they might have a problem metabolizing medications should seek answers from medical professionals - and be persistent, if necessary, Nelson said.
"Be your own advocate," she said.
Meanwhile, Daggett just celebrated her 70th birthday, a milestone she easily could not have lived to see.
Events Jan 19th 2020 - 6pm
Otter Tail County residents invited to story workshops
Traffic and Construction Jan 19th 2020 - 4pm
Plowing in downtown Fargo to begin Sunday evening; drivers urged to remove vehicles
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