\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{cite} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 93, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/93\hfil $p$-Kirchoff fractional equation] {Existence of solutions for fractional $p$-Kirchhoff equations with critical nonlinearities} \author[P. K. Mishra, K. Sreenadh \hfil EJDE-2015/93\hfilneg] {Pawan Kumar Mishra, Konijeti Sreenadh} \address{Pawan Kumar Mishra \newline Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi-16, India} \email{pawanmishra31284@gmail.com} \address{Konijeti Sreenadh \newline Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khaz, New Delhi-16, India} \email{sreenadh@gmail.com} \thanks{Submitted September 2, 2014. Published April 12, 2015.} \subjclass[2000]{34B27, 35J60, 35B05} \keywords{Kirchhoff non-local operators; fractional differential equations; \hfill\break\indent critical exponent} \begin{abstract} In this article, we show the existence of non-negative solutions of the fractional $p$-Kirchhoff problem \begin{gather*} -M(\int_{\mathbb{R}^{2n}} |u(x)-u(y)|^pK(x-y)dx\,dy)\mathcal{L}_Ku =\lambda f(x,u)+|u|^{p^* -2}u\quad \text{in }\Omega,\\ u=0\quad \text{in }\mathbb{R}^{n}\setminus\Omega, \end{gather*} where $\mathcal{L}_K$ is a $p$-fractional type non local operator with kernel $K$, $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, $M$ and $f$ are continuous functions, and $p^*$ is the fractional Sobolev exponent. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this work, we study the existence of solutions for the following $p$-Kirchhoff equation \begin{equation} \label{eMlambda} \begin{gathered} -M\Big( \int_{\mathbb{R}^{2n}} |u(x)-u(y)|^pK(x-y)dx\,dy\Big) \mathcal{L}_Ku =\lambda f(x,u)+|u|^{p^* -2}u \quad \text{in } \Omega,\\ u =0 \quad \text{in } \mathbb{R}^{n}\setminus\Omega, \end{gathered} \end{equation} where $p>1$, $n> ps$ with $s\in(0,1)$, $p^{*}=\frac{np}{n-ps}$, $\lambda$ is a positive parameter, $\Omega\subset\mathbb{R}^{n}$ is a bounded domain with smooth boundary and $M:\mathbb{R}^{+}\to \mathbb{R}^{+}$, $f:\overline{\Omega}\times \mathbb{R} \to \mathbb{R}$ are continuous functions that satisfy some growth assumptions which will be stated later. Here the operator $\mathcal{L}_K$ is the $p$-fractional type non-local operator defined as follows: \begin{equation*} \mathcal{L}_Ku(x)=2\int_{\mathbb{R}^{n}}|u(x)-u(y)|^{p-2}(u(x)-u(y))K(x-y)dy \quad \text{for all } x\in\mathbb{R}^n, \end{equation*} where $K:\mathbb{R}^{n}\setminus \{0\}\to(0,+\infty)$ is a measurable function with the property that \begin{equation}\label{K2} \parbox{10cm}{there exists $\theta >0$ and $s\in (0,1)$ such that $\theta|x|^{-(n+ps)}\leq K(x)\leq\theta^{-1}|x|^{-(n+ps)}$ for any $x\in\mathbb{R}^{n}\setminus\{0\}$.} \end{equation} It is immediate to observe that $mK\in L^{1}(\mathbb{R}^{n})$ by setting $m(x)=\min\{|x|^{p},1\}$. A typical example for $K$ is given by $K(x)=|x|^{-(n+ps)}$. In this case problem \eqref{eMlambda} becomes \begin{equation}\label{fraclapl} \begin{gathered} M\Big( \int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}dx\,dy \Big)(-\Delta)^{s}_p u =\lambda f(x,u)+|u|^{p^* -2}u\quad \text{in } \Omega,\\ u=0 \quad \text{in } \mathbb{R}^{n}\setminus \Omega, \end{gathered} \end{equation} where $(-\Delta)^{s}_p$ is the fractional $p$-Laplace operator defined as \[ - 2\int_{\mathbb{R}^n}\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\alpha}} dy. \] Problems \eqref{eMlambda} and \eqref{fraclapl} are variational in nature and the natural space to look for solutions is the fractional Sobolev space $W^{s,p}_{0}(\Omega)$ (see \cite{valpal}). To study \eqref{eMlambda} and \eqref{fraclapl}, it is important to encode the `boundary condition' $u=0$ in $\mathbb{R}^n\setminus\Omega$ (which is different from the classical case of the Laplacian) in the weak formulation. Also that in the norm $\|u\|_{W^{s,p}(\mathbb{R}^n)}$, the interaction between $\Omega$ and $\mathbb{R}^n\setminus\Omega$ gives positive contribution. Inspired by \cite{sv2, sv3}, we define the function space for $p$-case as \begin{align*} X=\Big\{&u:\mathbb{R}^n\to \mathbb{R}: u\text{ is measurable},\; u\big|_\Omega\in L^p(\Omega), \\\ & (u(x)-u(y))\sqrt[p]{K(x-y)}\in L^p(Q)\Big\}, \end{align*} where $Q:=\mathbb{R}^{2n}\setminus ({\mathcal C}\Omega\times{\mathcal C}\Omega)$. The space $X$ is endowed with a norm, defined as \begin{equation}\label{norma} \|u\|_{X}=\Big(\|u\|_{L^p(\Omega)}+\int_Q |u(x)-u(y)|^pK(x-y)dx\,dy\Big)^{1/p}\,. \end{equation} It is immediate to observe that bounded and Lipschitz functions belong to $X$, thus $X$ is not reduced to $\{0\}$. These spaces for the case $p=2$ are studied in \cite{sv2, sv3}. The function space $X_0$ denotes the closure of $C^{\infty}_{0}(\Omega)$ in $X$. By \cite[Lemma 4]{fiscella}, the space $X_0$ is a Banach space which can be endowed with the norm, defined as \begin{equation}\label{normaz} \|u\|_{X_0}=\Big(\int_Q |u(x)-u(y)|^p K(x-y)dx\,dy\Big)^{1/p}\,. \end{equation} Note that in \eqref{norma} and \eqref{normaz}, the integrals can be extended to all $\mathbb{R}^{2n}$, since $u=0$ a.e. in $\mathbb{R}^n \setminus\Omega$. In view of our problem, we assume that $M:\mathbb{R}^+ \to\mathbb{R}^+$ satisfies the following conditions: \begin{itemize} \item[(M1)] $M:\mathbb{R}^{+}\to \mathbb{R}^{+}$ is an increasing and continuous function. \item[(M2)] There exists $ m_0> 0$ such that $M(t)\geq m_0= M(0)$ for any $t\in\mathbb{R}^{+}$. \end{itemize} A typical example for $M$ is given by $M(t)=m_0 +tb$ with $b\geq 0$. Also, we assume that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a continuous function that satisfies: \begin{itemize} \item[(F1)] $f(x,t)=0$ for any $x\in\Omega$,\; $t\leq 0$ and $ \lim_{t \to 0} \frac{f(x,t)}{t^{p-1}}=0,$ uniformly in $x\in\Omega$; \item[(F2)] There exists $ q\in (p, p^*)$ such that $ \lim_{t \to \infty} \frac{f(x,t)}{t^{q-1}}=0$, uniformly in $x\in\Omega$; \item[(F3)] There exists $ \sigma\in (p, p^*)$ such that for any $x\in\Omega$ and $t>0$, \[ 0<\sigma F(x,t)=\sigma\int^{t}_{0}f(x,s)ds\leq tf(x,t). \] \end{itemize} \begin{definition} \label{def1.1} \rm A function $u\in X_0$ is called weak solution of \eqref{eMlambda} if $u$ satisfies \begin{equation} \begin{aligned} &M(\|u\|^p_{X_0})\int_{\mathbb{R}^{2n}} |u(x)-u(y)|^{p-2}(u(x)-u(y))(\varphi(x) -\varphi(y))K(x-y) dx\,dy \\ &= \lambda\int_\Omega f(x, u(x))\varphi(x)\,dx+\int_\Omega |u(x)|^{p^*-2}u(x)\varphi(x)dx \quad \forall\varphi \in X_0. \end{aligned} \label{wf} \end{equation} \end{definition} Thanks to our assumptions on $\Omega$, $M$, $f$ and $K$, all the integrals in \eqref{wf} are well defined if $u$, $\varphi\in X_0$. We also point out that the odd part of function $K$ gives no contribution to the integral of the left-hand side of \eqref{wf}. Therefore, it would be not restrictive to assume that $K$ is even. The fractional Laplacian $(-\Delta)_{2}^{s}$ operator has been a classical topic in Fourier analysis and nonlinear partial differential equations for a long time. Non-local operators, naturally arise in continuum mechanics, phase transition phenomena, population dynamics and game theory, see \cite{caf} and references therein. Fractional operators are also involved in financial mathematics, where Levy processes with jumps appear in modeling the asset prices (see \cite{app}.) In \cite{afev} author gave motivation for the study of fractional Kirchhoff equations occurring in vibrating strings. Here we study the $p$-fractional version of the problem studied in \cite{afev}. We follow and adopt the same approach as in \cite{afev} to obtain our results. Recently, much interest has grown to the study of critical exponent problem for non-local equations. The Brezis-Nirenberg problem for the Kirchhoff type equations are studied in \cite{acf, dn, gf} and references therein. Also, there are many works on the study of critical problems in a non-local setting inspired by fractional Laplacian \cite{ capella, gf, afev, sv1, sv2, sv3, tan}. Variational problems involving $p$-fractional operator with sub-critical and sign changing nonlinearities are studied in \cite{SS1,SS2}, using Nehari manifold and fibering maps. In \cite{afev}, authors considered the fractional Kirchhoff problem \begin{equation} \label{eLlambda} \begin{gathered} -M\Big( \int_{\mathbb{R}^{2n}} |u(x)-u(y)|^2 K(x-y)dx\,dy\Big) \mathcal{L}_K u =\lambda f(x,u)+|u|^{2^* -2}u \quad \text{in } \Omega,\\ u =0 \quad \text{in } \mathbb{R}^{n}\setminus\Omega, \end{gathered} \end{equation} with $K(x)\sim |x|^{-(n+2s)}$ and $f(x,u)$ having sub-critical growth. Using mountain pass Lemma and the study of compactness of Palais-Smale sequences, they established the existence of solutions of \eqref{eLlambda} for large $\lambda$. Inspired by the above articles, in this paper we will investigate the existence of a nontrivial solution for $p$-fractional Kirchhoff problem stated in \eqref{eMlambda}. To the best of our knowledge, there are no works on $p$-Kirchhoff fractional equations. With this introduction, we state our main result. \begin{theorem} \label{thm1} Let $s\in(0,1)$, $p>1$, $n> ps$ and $\Omega$ be a bounded open subset of $\mathbb{R}^{n}$. Assume that the functions $K(x)$, $M(t)$ and $f(x,t)$ satisfy conditions \eqref{K2}, {\rm (M1)--(M2)} and {\rm (F1)--(F3)}. Then there exists $\lambda^*>0$ such that problem \eqref{eMlambda} has a nontrivial solution $u_\lambda$ for all $\lambda\geq\lambda^*$. Moreover, $ \lim_{\lambda\to\infty}\|u_\lambda\|_{X_0}=0$. \end{theorem} \section{Auxiliary problem and variational formulation} To prove Theorem \ref{thm1}, we first study an auxiliary truncated problem. Given $\sigma$ as in (F3) and $a\in\mathbb{R}$ such that $m_00$ such that $M(t_0)=a$. Now, by setting \begin{equation}\label{trun} M_a(t):= \begin{cases} M(t) & \text{if } 0\leq t\leq t_0,\\ a & \text{if } t\geq t_0, \end{cases} \end{equation} we introduce the auxiliary problem \begin{equation} \begin{gathered} -M_a(\|u\|^{p}_{X_0})\mathcal{L}_K u =\lambda f(x,u)+|u|^{p^* -2}u \quad \text{in } \Omega,\\ u=0\quad \text{in } \mathbb{R}^{n}\setminus\Omega, \end{gathered}\label{Pa} \end{equation} with $f$ satisfying conditions (F1)--(F3) and $\lambda$ being a positive parameter. By (M1), we also note that \begin{equation}\label{ma} M_a(t)\leq a\quad\text{for all } t\geq 0. \end{equation} We obtain the following result. \begin{theorem} \label{thm2} Assume that $K(x)$, $M(t)$ and $f(x,t)$ satisfies \eqref{K2}, {\rm (M1)--(M2)} and {\rm (F1)--(F3)}, respectively. Then there exists $\lambda_0 >0$ such that problem \eqref{Pa} has a nontrivial weak solution, for all $\lambda\geq\lambda_0$ and for all $a\in \big(m_0, \frac{\sigma}{p}m_0\big)$. \end{theorem} For the proof of Theorem~\ref{thm2}, we observe that problem~\eqref{Pa} has a variational structure. The Euler functional corresponding to \eqref{Pa} is $\mathcal{J}_{a,\lambda}:X_0\to \mathbb{R}$ defined as follows $$ \mathcal{J}_{a,\lambda}(u)=\frac{1}{p}\widehat{M_a} (\|u\|^{p}_{X_0})-\lambda\int_\Omega F(x, u(x))dx -\frac{1}{p^*}\int_\Omega |u(x)|^{p^*}dx, $$ where \[ \widehat{M_a}(t)=\int^{t}_{0}M_a(s)ds. \] Then the functional $\mathcal{J}_{a,\lambda}$ is Fr\'echet differentiable on $X_0$ and for any $\varphi\in X_0$, \begin{equation}\label{derivata} \begin{aligned} &\langle \mathcal{J}'_{a,\lambda}(u), \varphi\rangle \\ &= M_a(\|u\|^{p}_{X_0})\int_Q |u(x)-u(y)|^{p-2} \big(u(x)-u(y)\big)\big(\varphi(x)-\varphi(y)\big)K(x-y)\,dx\,dy\\ &\quad -\lambda\int_\Omega f(x, u(x))\varphi(x)\,dx -\int_\Omega |u(x)|^{p^*-2}u(x)\varphi(x)dx\,. \end{aligned} \end{equation} Now we prove that the functional $\mathcal{J}_{a,\lambda}$ has the geometric features required by the Mountain Pass Theorem. \begin{lemma}\label{mp1} Let $K(x)$, $M(t)$ and $f(x,t)$ be three functions satisfying \eqref{K2}, {\rm (M1)--(M2)} and {\rm (F1)--(F3)}, respectively. Then there exist two positive constants $\rho$ and $\alpha$ such that \begin{equation} \mathcal{J}_{a,\lambda}(u)\geq\alpha>0, \end{equation} for any $u\in X_0$ with $\|u\|_{X_0}=\rho$. \end{lemma} \begin{proof} By (F1) and (F2), it follows that, for any $\epsilon>0$ there exists $\delta=\delta(\epsilon)>0$ such that \begin{equation}\label{thebre} |F(x,t)|\leq \epsilon|t|^p +\delta|t|^q\,. \end{equation} By (M2) and \eqref{thebre}, we obtain \[ \mathcal{J}_{a,\lambda}(u)\geq \frac{m_0}{p}\|u\|^{p}_{X_0} -\epsilon\lambda\int_\Omega |u(x)|^p dx -\delta\lambda\int_\Omega|u(x)|^q dx-\frac{1}{p^*}\int_\Omega|u(x)|^{p^*} dx. \] So, by fractional Sobolev inequality (see \cite[Theorem 6.5]{valpal}), there is a positive constant $C=C(\Omega)$ such that \begin{align*} \mathcal{J}_{a,\lambda}(u) \geq \big(\frac{m_0}{p}-\epsilon\lambda C\big) \|u\|^{p}_{X_0}-\delta\lambda C\|u\|^{q}_{X_0}-C\|u\|^{p^*}_{X_0}. \end{align*} Therefore, by fixing $\epsilon$ such that $ \frac{m_0}{p}-\epsilon\lambda C >0$, since $p\rho$. \end{lemma} \begin{proof} We fix $u_0\in X_0$ such that $\|u_0\|_{X_0}=1$ and $u_0\geq 0$ a.e. in $\mathbb{R}^n$. For $t>0$, by $(F3)$ and \eqref{ma}, we obtain \[ \mathcal{J}_{a,\lambda}(tu_0)\leq a\frac{t^p}{p} -c_1t^\sigma \lambda\int_\Omega |u_0(x)|^\sigma dx+c_2 |\Omega| -\frac{t^{p^*}}{p^*}\int_\Omega |u_0(x)|^{p^*}dx. \] Since $\sigma>p$, passing to the limit as $t\to +\infty$, we obtain that $\mathcal{J}_{a,\lambda}(tu_0)\to -\infty$, so that the assertion follows by taking $e=t_{*}u_0$, with $t_* >0$ large enough. \end{proof} Now, we prove that the Palais-Smale sequence is bounded. \begin{lemma}\label{psb} Let $K(x)$, $M(t)$ and $f(x,t)$ be three functions satisfying \eqref{K2}, {\rm (M1)--(M2)} and {\rm (F1)--(F3)}, respectively. Let $\{u_{j}\}_{j\in\mathbb{N}}$ be a sequence in $X_0$ such that, for any $c\in(0,\infty)$, \begin{equation}\label{ps1} \mathcal{J}_{a,\lambda}(u_{j})\to c, \quad \mathcal{J}'_{a,\lambda}(u_{j})\to 0, \end{equation} as $j\to +\infty$. Then $\{u_j\}_{j\in\mathbb{N}}$ is bounded in $X_0$. \end{lemma} \begin{proof} By \eqref{ps1}, there exists $C>0$ such that \begin{equation}\label{4.1} |\mathcal{J}_{a,\lambda}(u_j)|\leq C,\quad \langle \mathcal{J}'_{a,\lambda}(u_j),u_j\rangle \leq C\|u_j\|_{X_0}, \end{equation} for any $j\in\mathbb{N}$. Moreover, by (M2), (F3), and \eqref{ma} it follows that \begin{equation} \label{4.2} \begin{aligned} \mathcal{J}_{a,\lambda}(u_j)-\frac{1}{\sigma}\mathcal{J}'_{a,\lambda}(u_j)(u_j) &\geq\frac{1}{p}\widehat{M_a}(\|u_j\|_{X_0}^{p})-\frac{1}{\sigma} M_a(\|u_j\|_{X_0}^{p})\|u_j\|_{X_0}^{p} \\ &\geq\big(\frac{1}{p}m_0-\frac{1}{\sigma}a\big)\|u_j\|_{X_0}^{p}. \end{aligned} \end{equation} On the other hand, from \eqref{4.1}, we obtain \begin{equation}\label{4.3} \mathcal{J}_{a,\lambda}(u_j)-\frac{1}{\sigma} \langle\mathcal{J}'_{a,\lambda}(u_j)(u_j)\rangle \leq C(1+\|u_j\|_{X_0}). \end{equation} Now, from \eqref{4.2} and \eqref{4.3} along with the assumption, $m_00, \end{equation} where \[ \Gamma:=\{\gamma\in C([0,1],\,X_0): \gamma(0)=0,\; \mathcal{J}_{a,\lambda}(\gamma(1))<0\}. \] The following result is needed to study the asymptotic behavior of the solution of problem \eqref{wf}. \begin{lemma}\label{infinito} Let $K(x)$, $M(t)$ and $f(x,t)$ be three functions satisfying \eqref{K2}, {\rm (M1)--(M2)} and {\rm (F1)--(F3)}. Then $\lim_{\lambda\to+\infty}c_{a,\lambda}=0$. \end{lemma} \begin{proof} Let $e\in X_0$ be the function given by Lemma \ref{mp2} and let $\{\lambda_j\}_{j\in\mathbb{N}}$ be a sequence such that $\lambda_j\to+\infty$. Since $\mathcal{J}_{a,\lambda}$ satisfies the Mountain Pass geometry, it follows that there exists $t_\lambda>0$ such that $\mathcal{J}_{a,\lambda}(t_\lambda e)= \max_{t\geq 0} \mathcal{J}_{a,\lambda}(te)$. Hence, $\langle \mathcal{J}'_{a,\lambda}(t_\lambda e), e\rangle=0$ and by \eqref{derivata}, we obtain \begin{equation}\label{3.1} t_\lambda^{p-1} \|e\|^p_{X_0} M_a(t^{p}_{\lambda}\|e\|^p_{X_0})= \lambda\int_\Omega f(x, t_\lambda e(x))e(x)\,dx +t^{p^*-1}_{\lambda}\int_\Omega |e(x)|^{p^*}dx\,. \end{equation} Now, by construction $e\geq0$ a.e. in $\mathbb{R}^n$. So, by (F3), \eqref{ma} and \eqref{3.1} it follows that \[ a\|e\|^p_{X_0}\geq t^{p^*-p}_{\lambda}\int_\Omega |e(x)|^{p^*}dx, \] which implies that $t_\lambda$ is bounded for any $\lambda>0$. Thus, there exists $\beta\geq 0$ such that $t_{\lambda_j}\to \beta$ as $j\to+\infty$. So, by \eqref{ma} and \eqref{3.1} there exists $D>0$ such that \begin{equation}\label{D} \lambda_j\int_\Omega f(x, t_{\lambda_j} e(x))e(x)\,dx +t^{p^*-1}_{\lambda_j}\int_\Omega |e(x)|^{p^*}dx =t_{\lambda_j}^{p-1}M_a(t^{p}_{\lambda_j}\|e\|^p_{X_0})\leq D, \end{equation} for any $j\in\mathbb{N}$. We claim that $\beta=0$. Indeed, if $\beta>0$ then by $(F1)$, $(F2)$, for any $\epsilon>0$, there exists $\delta=\delta(\epsilon)>0$ such that \begin{equation*} |f(x,t)|\leq \epsilon|t|^{p-1} +q\delta|t|^{q-1}\quad \text{for all } t\in \mathbb{R}, \end{equation*} and so, by the Dominated Convergence Theorem, \[ \int_\Omega f(x, t_{\lambda_j} e(x))e(x)\,dx \to\int_\Omega f(x, \beta e(x))e(x)\,dx\quad\text{as }j\to+\infty. \] Now, since $\lambda_j\to+\infty$, we obtain \[ \lim_{j\to+\infty}\lambda_j\int_\Omega f(x, t_{\lambda_j} e(x))e(x)\,dx +t^{p^*-1}_{\lambda_j}\int_\Omega |e(x)|^{p^*}dx=+\infty, \] which contradicts \eqref{D}. Thus, we have that $\beta=0$. Now, we consider the following path $\gamma_*(t)=te$ for $t\in[0,1]$ which belongs to $\Gamma$. Using $(F3)$, we obtain \begin{equation}\label{3.2} 0 \delta\}} |u(x)|^p |x-y|^{-n-ps}\,dx\,dy=0. \end{gather} \end{proposition} \begin{proof} We set $ \zeta_\delta:=\big( \int_{B_\delta(\xi)} |u(x)|^{p^*}\,dx\big)^{p/p^*}$ and we remark that \begin{equation}\label{B796} \lim_{\delta\to0} \zeta_\delta=0. \end{equation} Also we observe that, using the H\"older's inequality with exponents $ \frac{p^*}{p}=\frac{n}{n-ps}$ and $ \frac{n}{ps}$, we obtain \begin{equation}\label{B7} \int_{B_\delta(\xi)} |u(x)|^p\,dx\le \Big( \int_{B_\delta(\xi)} |u(x)|^{p^*}\,dx\Big)^{p/p^*} \Big( \int_{B_\delta(\xi)} 1\,dx\Big)^{ps/n} \le C \zeta_\delta \delta^{ps}, \end{equation} for some $C>0$ independent of $\delta$ (in what follows we will possibly change $C$ from line to line). Moreover \begin{equation}\label{eqB6} (U\times V)\cap \{|x-y|\le\delta\}\,\subseteq\, B_{2\delta}(\xi)\times B_{2\delta}(\xi). \end{equation} Indeed, if $(x,y)\in U\times V=B_\delta(\xi)\times \mathbb{R}^n$, with $|x-y|\le\delta$, we obtain $ |\xi-y|\le |\xi-x|+|x-y|\le \delta+\delta,$ and so we obtain \eqref{eqB6}. On the other hand, if $(x,y)\in U\times V=\mathbb{R}^n\times B_\delta(\xi)$ with $|x-y|\le\delta$, we obtain $$ |\xi-x|\le |\xi-y|+|y-x|\le\delta+\delta,$$ and this completes the proof of \eqref{eqB6}. Now using the change of variable $z:=x-y$ and using \eqref{eqB6}, we obtain \begin{align*} &\int_{x\in U}\int_{y\in V\cap\{|x-y|\le \delta\}} |u(x)|^p |x-y|^{p-n-ps}\,dx\,dy\\ &\le \int_{x\in B_{2\delta}(p)}\int_{y\in B_{2\delta}(p)\cap\{|x-y|\le \delta\}} |u(x)|^p |x-y|^{p-n-ps}\,dx\,dy \\ & \le \int_{x\in B_{2\delta}(\xi)}\int_{z\in B_{\delta}} |u(x)|^p |z|^{p-n-ps}\,dx\,dz\\ &\le C\delta^{p-ps} \int_{x\in B_{2\delta}(\xi)}|u(x)|^p \,dx. \end{align*} Using this and \eqref{B7}, we obtain \begin{equation}\label{dell} \begin{aligned} &\delta^{-p}\int_U\int_{V\cap\{|x-y|\le \delta\}} |u(x)|^p |x-y|^{p-n-ps}\,dx\,dy\\ &\leq C\delta^{-ps} \int_{x\in B_{2\delta}(\xi)}|u(x)|^p \,dx \le C\zeta_\delta. \end{aligned} \end{equation} So, \eqref{dell} and \eqref{B796} imply \eqref{EX.1}. Now, we prove \eqref{EX.2}. For this, we fix an auxiliary parameter $K>2$ (such parameter will be taken arbitrarily large at the end, after taking $\delta\to0$). We observe that \begin{equation}\label{BBeqB} U\times V\,\subseteq\, \big(B_{K\delta}(\xi)\times\mathbb{R}^n\big) \cup\big( (\mathbb{R}^n\setminus B_{K\delta}(\xi)) \times B_{\delta}(\xi \big)\big). \end{equation} Indeed, if $U\times V=B_\delta(\xi)\times \mathbb{R}^n$, then of course $U\times V\subseteq B_{K\delta}(\xi)\times\mathbb{R}^n$, hence \eqref{BBeqB} is obvious. If instead $(x,y)\in U\times V=\mathbb{R}^n\times B_\delta(\xi)$, we distinguish two cases: if $x\in B_{K\delta}(\xi)$ then $(x,y)\in B_{K\delta}(\xi)\times \mathbb{R}^n$; if $x\in\mathbb{R}^n\setminus B_{K\delta}(\xi)$, then \[ (x,y)\in (\mathbb{R}^n\setminus B_{K\delta}(\xi))\times V= (\mathbb{R}^n\setminus B_{K\delta}(\xi))\times B_\delta(\xi). \] This completes the proof of \eqref{BBeqB}. Now, we compute \begin{equation} \label{56.1} \begin{aligned} &\int_{x\in B_{K\delta}(\xi)}\int_{y\in\mathbb{R}^n\cap\{|x-y|> \delta\}} |u(x)|^p |x-y|^{-n-ps}\,dx\,dy\\ &= \int_{x\in B_{K\delta}(\xi)}\int_{z\in\mathbb{R}^n\setminus B_\delta} |u(x)|^p |z|^{-n-ps}\,dx\,dz \\ &= C\delta^{-ps} \int_{x\in B_{K\delta}(\xi)} |u(x)|^p \,dx \le C\zeta_{K\delta}, \end{aligned} \end{equation} where \eqref{B7} has been used again in the last step. Now, we observe that if $x\in\mathbb{R}^n\setminus B_{K\delta}(\xi)$ and $y\in B_\delta(\xi)$ then \begin{align*} |x-y| &\ge |x-\xi|-|y-\xi|=\frac{|x-\xi|}2+\frac{|x-\xi|}2-|y-p|\\ &\ge \frac{|x-\xi|}2+\frac{K\delta}2-\delta\ge\frac{|x-\xi|}2.\end{align*} As a consequence we infer that \begin{align*} &\int_{x\in \mathbb{R}^n\setminus B_{K\delta}(\xi)} \int_{y\in B_\delta(\xi)}|u(x)|^p |x-y|^{-n-ps}\,dx\,dy \\ & \le C \int_{x\in \mathbb{R}^n\setminus B_{K\delta}(\xi)} \int_{y\in B_\delta(\xi)} |u(x)|^p |x-\xi|^{-n-ps}\,dx\,dy \\ &= C\delta^n \int_{x\in \mathbb{R}^n\setminus B_{K\delta}(p)}|u(x)|^p |x-\xi|^{-n-ps}\,dx. \end{align*} Now using the H\"older's inequality with exponents $ \frac{p^*}{p}=\frac{n}{n-ps}$ and $ \frac{n}{ps}$, we obtain \begin{equation} \label{56.2.8} \begin{aligned} &\int_{x\in \mathbb{R}^n\setminus B_{K\delta}(\xi)} \int_{y\in B_\delta(\xi)}|u(x)|^p |x-y|^{-n-ps}\,dx\,dy \\ &\le C\delta^n \Big( \int_{x\in \mathbb{R}^n\setminus B_{K\delta}(\xi)}|u(x)|^{p^*}\,dx\Big)^{p/p^*} \Big(\int_{x\in \mathbb{R}^n\setminus B_{K\delta}(\xi)} |x-\xi|^{-(n+ps)n/ps}\,dx\Big)^{ps/n} \\ &\leq C\delta^n \| u\|^p_{L^{p^*}(\mathbb{R}^n)} \Big(\int_{K\delta}^{+\infty} \rho^{-((n+ps)n/ps)+(n-1)}d\rho \Big)^{ps/n} \\ &= C\delta^n \| u\|^p_{L^{p^*}(\mathbb{R}^n)} \big((K\delta)^{-n^2/ps}\big)^{ps/n}\\ &= CK^{-n}\| u\|^p_{L^{p^*}(\mathbb{R}^n)}. \end{aligned} \end{equation} By collecting the results in \eqref{BBeqB}, \eqref{56.1} and \eqref{56.2.8}, we obtain \begin{align*} &\int_U\int_{V\cap\{|x-y|> \delta\}} |u(x)|^p |x-y|^{-n-ps}\,dx\,dy\\ &\le \int_{x\in B_{K\delta}(\xi)}\int_{y\in\mathbb{R}^n\cap\{|x-y|> \delta\}} |u(x)|^p |x-y|^{-n-ps}\,dx\,dy \\ &\quad+ \int_{x\in \mathbb{R}^n\setminus B_{K\delta}(\xi)} \int_{y\in B_\delta(\xi)} |u(x)|^p |x-y|^{-n-ps}\,dx\,dy\\ &\le C\zeta_{K\delta} +CK^{-n}\| u\|^p_{L^{p^*}(\mathbb{R}^n)}. \end{align*} From this, we first take $\delta\to0$ and then $K\to+\infty$ to obtain \eqref{EX.2} (using again \eqref{B796}). \end{proof} \section{Proofs of Theorems \ref{thm1} and \ref{thm2}} We need the following lemma in which we study the local Palais-Smale sequences and show the Palais-Smale condition, $(PS)_c$ in short, below the first critical level. \begin{lemma}\label{lemm} There exists $\lambda_0>0$ such that $\mathcal{J}_{a,\lambda}$ satisfies $(PS)_{c_{a,\lambda}}$ for all $\lambda>\lambda_0$, where $c_{a,\lambda}$ is defined in \eqref{calam}. \end{lemma} \begin{proof} Let $\{u_j\}$ be a Palais-Smale sequence in $X_0$ at level $c_{a,\lambda}$ i.e. $\{u_j\}$ satisfies \eqref{ps1}. By lemma \ref{psb}, $\{u_j\}$ is bounded in $X_0$ and so upto subsequence $\{u_j\}$ converges weakly to $u$ in $X_0$, strongly in $L^q$ for all $1\leq q< p^*$ and point wise to u almost everywhere in $\Omega$. Also there exists $h\in L^p(\Omega)$ such that $|u_j(x)|\leq h(x) $ a.e. in $\Omega$. Also $\{\|u_j\|_{X_0}\}$ as a real sequence converges to $\alpha$ (say). Since $M_a$ is continuous, $M_a(\|u_j\|^p_{X_0})\to M_a(\alpha^p)$. Now we claim that \begin{equation} \label{claim} \|u_j\|^{p}_{X_0}\to \|u\|^{p}_{X_0}\quad\text{as }j\to+\infty, \end{equation} Once the claim is proved, we can invoke Brezis-Leib lemma to prove that $u_j$ converges to $u$ strongly in $X_0$. We know that $\{u_j\}$ is also bounded in $W_0^{s,p}(\Omega)$. So we may assume that there exists two positive measures $\mu$ and $\nu$ on $\mathbb{R}^n$ such that \begin{equation}\label{prkh1} |(-\Delta)^s_p u_j|^p dx \stackrel{*}{\rightharpoonup}\mu\quad\text{and}\quad|u_j|^{p^*}\rightharpoonup\nu, \end{equation} in the sense of measure. Moreover, (see, \cite{pal}), we have a countable index set $J$, positive constants $\{\nu_j\}_{j\in J}$ and $\{\mu_j\}_{j\in J}$ such that \begin{gather} \nu=|u|^{p^*}dx+\sum_{i\in J} \nu_i\delta_{x_i}, \\ \label{prkh2} \mu\geq|(-\Delta)^s_p u|^p dx+\sum_{i\in J} \mu_i\delta_{x_i},\quad \nu_i\leq S\mu^{p^*/p}_{i}, \end{gather} where $S$ is the best constant of the embedding $W^{s,p}_{0}(\Omega)$ into $L^{p^*}(\Omega)$. Our goal is to show that $J$ is empty. Suppose not, then there exists $i\in J$. For this $x_i$, define $\phi_\delta^i(x)=\phi(\frac{x-x_i}{\delta}), x\in \mathbb{R}^n$ and $\phi \in C_0^\infty(\mathbb{R}^n,[0,1])$ such that $\phi=1$ in $B(0,1)$ and $\phi=0$ in $\mathbb{R}^n\setminus B(0,2)$. Since $\{\phi_\delta^iu_j\}$ is bounded in $X_0$, we have $\mathcal{J}'_{a,\lambda}(u_j)(\phi_\delta^i u_j)\to 0$ as $j\to+\infty$. That is, \begin{equation} \label{3.27} \begin{aligned} &M_a(\|u_j\|^{p}_{X_0})\int_{\mathbb{R}^{2n}} u_j(x) |u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y))\\ &\quad\times \big(\phi_\delta^i(x)-\phi_\delta^i(y)\big)K(x-y)\,dx\,dy \\ &=-M_a(\|u_j\|^{p}_{X_0})\int_{\mathbb{R}^{2n}} \phi_\delta^i(y)|u_j(x) -u_j(y)|^pK(x-y)\,dx\,dy \\ &\quad+\lambda\int_\Omega f(x, u_j(x))\phi_\delta^i(x)u_j(x)dx +\int_\Omega |u_j(x)|^{p^*}\phi_\delta^i(x)dx\,+o_j(1), \end{aligned} \end{equation} as $j\to\infty$. Now using H\"older's inequality and the fact that $\{u_j\}$ is bounded in $X_0$, we obtain \begin{align*} &\big|\int_{\mathbb{R}^{2n}}u_j(x)|(u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y)) \big(\phi_\delta^i(x)-\phi_\delta^i(y)\big)K(x-y)\,dx\,dy\big| \\ &\leq C \Big(\int_{\mathbb{R}^{2n}} |u_j(x)|^p|\phi_\delta^i(x) -\phi_\delta^i(y)|^pK(x-y)\,dx\,dy\Big)^{1/p}. \end{align*} Now we claim that \begin{equation} \lim_{\delta\to 0}\Big[\lim_{j\to+\infty}\Big(\int_{\mathbb{R}^{2n}} |u_j(x)|^p|\phi_\delta^i(x)-\phi_\delta^i(y)|^pK(x-y)\,dx\,dy\Big)\Big]=0. \end{equation} Using the Lipschitz regularity of $\phi_\delta^i$, we have, for some $L\ge 0$, \begin{equation} \label{facl} \begin{aligned} &\int_{\mathbb{R}^{2n}} |u_j(x)|^p |\phi_{\delta}^{i}(x) -\phi_{\delta}^{i}(y)|^p K(x-y)\,dx\,dy\\ &\leq\frac{1}{\theta}\int_{\mathbb{R}^{2n}} |u_j(x)|^p|\phi_\delta^i(x) -\phi_\delta^i(y)|^p|x-y|^{-n-ps}\,dx\,dy\\ &\leq \frac{L^p\delta^{-p}}{\theta}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n\cap\{|x-y| \leq \delta\}} |u_j(x)|^p|x-y|^{p-n-ps}\,dx\,dy\\ &\quad+\frac{2^p}{\theta}\int_{\mathbb{R}^n}\int_{\mathbb{R}^n\cap\{|x-y|>\delta\}} |u_j(x)|^p|x-y|^{-n-ps}\,dx\,dy\\ &\leq C\frac{(L^p\delta^{-p}+2^p)}{\theta}\int_{\mathbb{R}^n}|h(x)|^p\,dx\,dy <+\infty, \end{aligned} \end{equation} with $C=C(n,s,\delta)>0$. So, by dominated convergence theorem \begin{align*} &\lim_{j\to+\infty}\int_{\mathbb{R}^{2n}} |u_j(x)|^p |\phi_\delta^i(x)-\phi_\delta^i(y)|^p K(x-y)\,dx\,dy \\ &=\int_{\mathbb{R}^{2n}} |u(x)|^p|\phi_\delta^i(x)-\phi_\delta^i(y)|^p K(x-y)\,dx\,dy. \end{align*} Now, following the calculations in \eqref{facl}, we obtain \begin{equation} \begin{aligned} &\int_{U\times V} |u(x)|^p |\phi_\delta^i(x)-\phi_\delta^i(y)|^p K(x-y)\,dx\,dy \\ &\leq\frac{L^p}{\theta}\delta^{-p}\int_{U}\int_{V\cap\{|x-y|\leq \delta\}} |u(x)|^p |x-y|^{p-n-ps}\,dx\,dy \\ &\quad+\frac{2^p}{\theta}\int_{U}\int_{V\cap\{|x-y|>\delta\}} |u(x)|^p|x-y|^{-n-ps}\,dx\,dy, \end{aligned} \end{equation} where $U$ and $V$ are two generic subsets of $\mathbb{R}^n$. Next we claim that \[ \int_{\mathbb{R}^{2n}} |u(x)|^p|\phi_\delta^i(x)-\phi_\delta^i(y)|^p K(x-y)\,dx\,dy\to 0,\quad \text{as } \delta \to 0. \] When $U=V=\mathbb{R}^n\setminus B(x_i,\delta)$ claim follows. When $U\times V=B(x_i,\delta)\times\mathbb{R}^n$ and $U\times V=\mathbb{R}^n\times B(x_i,\delta)$, we can use Proposition \ref{uperv} to prove the claim. Thus \begin{equation} \lim_{\delta\to 0}\int_{\mathbb{R}^{2n}} |u(x)|^p|\phi_\delta^i(x)-\phi_\delta^i(y)|^p K(x-y)\,dx\,dy=0. \end{equation} Hence \begin{equation}\label{3.34} \begin{aligned} &M_a (\|u_j\|^{p}_{X_0})\int_{\mathbb{R}^{2n}} u_j(x)| (u_j(x)-u_j(y)|^{p-2}(u_j(x)-u_j(y))\\ &\times \big(\phi_\delta^i(x)-\phi_\delta^i(y)\big)K(x-y)\,dx\,dy\to 0, \end{aligned} \end{equation} as $\delta \to 0$ and $j\to \infty$. Now, using H\"older's inequality, \begin{equation} \label{reqe} \begin{aligned} &\Big|\int_{\mathbb{R}^n}\frac{u_j(x)-u_j(y)} {|x-y|^{n+ps}}\Big|^p\\ &\leq 2^{p-1}\Big[|u_j(y)|^p\Big|\int_{\mathbb{R}^n\setminus\Omega} \frac{1}{|x-y|^{n+ps}}\Big|^p +\Big|\int_{\Omega}\frac{u_j(x)-u_j(y)}{|x-y|^{n+ps}}dx\Big|^p\Big]\\ &\leq C_1|u_j(y)|^p+C_2\int_{\Omega}|u_j(x)-u_j(y)|^pK(x-y)dx, \end{aligned} \end{equation} where $C_1=2^{p-1}|\int_{\mathbb{R}^n\setminus\Omega}\frac{dx}{|x-y|^{n+ps}}|^p$ and $C_2=2^{p-1}/\theta$. Now using equations \eqref{reqe} and \eqref{prkh1}, we obtain \begin{equation} \label{3.36} \begin{aligned} &\liminf_{j\to+\infty} \int_{\mathbb{R}^n} \phi_\delta^i(y) \int_\Omega|u_j(x)-u_j(y)|^p K(x-y)\,dx\,dy \\ &\geq C_3\frac{1}{c(n,s)}\liminf_{j\to+\infty} \int_{\mathbb{R}^n} \phi_\delta^i(y) c(n,s) \Big|\int_{\mathbb{R}^n}\frac{u_j(x)-u_j(y)}{|x-y|^{n+ps}}\,dx\Big|^p dy \\ &\quad-C_4\liminf_{j\to+\infty}\int_{\mathbb{R}^n} \phi_\delta^i(y)|u_j(y)|^p dy \\ &\geq C_3\frac{1}{c(n,s)}\int_{\mathbb{R}^n} \phi_\delta^i(y)d\mu -C_4\int_{B(x_i,\delta)} |u(y)|^p dy, \end{aligned} \end{equation} where $C_3=1/C_2$ and $C_4=C_1/C_2$. Moreover, for a given $\epsilon >0$ there exist $C_\epsilon>0$ such that \begin{equation} |f(x,t)|\leq \epsilon|t|^{p-1} +C_\epsilon|t|^{q-1}\,. \end{equation} So, using Vitali's convergence theorem, we obtain \begin{equation}\label{3.38} \int_{B(x_i,\delta)} f(x, u_j(x))u_j(x)\phi_\delta^i(x)dx \to\int_{B(x_i,\delta)} f(x, u(x))u(x)\phi_\delta^i(x)dx, \end{equation} as $j\to+\infty$. We also observe that the integral goes to 0 as $\delta\to 0$. So, using \eqref{3.34}, \eqref{3.36}, \eqref{3.38} and \eqref{prkh1} in \eqref{3.27}, we obtain \begin{align*} &\int_\Omega \phi_\delta^i(x)d\nu+\lambda\int_{B(x_i,\delta)} f(x, u(x))u(x)\phi_\delta^i(x)dx\\ &\geq M_a(\alpha^p)C\Big(\int_\Omega\phi_\delta^i(y)d\mu -\int_{B(x_i,\delta)}|u(y)|^p dy\Big)+o_\delta(1). \end{align*} Now, by taking $\delta\to 0$, we conclude that $\nu_i\geq M_a(\alpha^p)C\mu_i\geq m_0 C\mu_i$. Then by \eqref{prkh2}, we obtain \begin{equation}\label{4.6} \nu_i\geq \frac{(m_0 C)^{n/ps}}{S^{(n-ps)/ps}}, \end{equation} for any $i\in J$, where $C=\frac{C_3}{c(n,s)}$, independent of $\lambda$. We will prove that \eqref{4.6} is not possible.\\ Consider \begin{equation}\label{4.7} \lim_{j\to+\infty}\Big(\mathcal{J}_{a,\lambda}(u_j)-\frac{1}{\sigma} \mathcal{J}'_{a,\lambda}(u_j)(u_j)\Big)=c_{a,\lambda}. \end{equation} Since, $m_00, \end{equation*} for all $\lambda$, but from Lemma \ref{infinito}, there exists $\lambda_0>0$ such that \[ c_{a,\lambda}<(\frac{1}{\sigma}-\frac{1}{p^*} )\frac{(m_0C)^{\frac{n}{ps}}}{S^\frac{n-ps}{ps}} \] for all $\lambda>\lambda_0$, which is a contradiction. Therefore $\nu_i=0$ or all $i\in J $. Hence $J$ is empty. Which implies $u_j\to u$ in $L^{p^*}(\Omega)$. So, by $\eqref{ps1}$ taking $\phi = u_j $ and using dominated convergence theorem, \begin{equation}\label{3.41} \lim_{j\to+\infty}M_a(\|u_j\|_{X_0}^{p})\|u_j\|_{X_0}^{p} =\lambda\int_\Omega f(x,u(x))u(x)dx+\int_\Omega|u(x)|^{2^*}dx. \end{equation} Now, we take $\phi=u$ in $\eqref{ps1}$ and recalling that $M_a(\|u_j\|^{p}_{X_0})\to M_a(\alpha^p)$, we obtain \begin{equation}\label{4.10} M_a(\alpha^p)\|u_j\|_{X_0}^{p} =\lambda\int_\Omega f(x, u(x))u(x)\,dx-\int_\Omega |u(x)|^{2^*}dx\,. \end{equation} So, combining \eqref{3.41} and \eqref{4.10}, we obtain \[ M_a(\|u_j\|_{X_0}^{p})\|u_j\|_{X_0}^{p}\to M_a(\alpha^p)\|u\|_{X_0}^{p},\quad \text{as }j\to+\infty. \] So, using this result, we have \begin{align*} M_a(\|u_j\|_{X_0}^{p})(\|u_j\|_{X_0}^{p}-\|u\|_{X_0}^{p}) &= M_a(\|u_j\|_{X_0}^{p})\|u_j\|_{X_0}^{p} -M_a(\alpha^p)\|u\|_{X_0}^{p} \\ &\quad -M_a(\|u_j\|_{X_0}^{p})\|u\|_{X_0}^{p}+M_a(\alpha^p)\|u\|_{X_0}^{p}, \end{align*} which leads to \[ M_a(\|u_j\|_{X_0}^{p})(\|u_j\|_{X_0}^{p}-\|u\|_{X_0}^{p})\to 0. \] Also \begin{equation} m_0(\|u_j\|_{X_0}^{p}-\|u\|_{X_0}^{p}) \leq M_a(\|u_j\|_{X_0}^{p})(\|u_j\|_{X_0}^{p}-\|u\|_{X_0}^{p}), \end{equation} which implies $\|u_j\|_{X_0}^{p}\to \|u\|_{X_0}^{p}$ and the claim is proved. Hence $u_j\to u$ strongly in $X_0$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2}] Using Lemma \ref{lemm} and by Mountain Pass Theorem, we obtain a critical point $u$ for the functional $\mathcal{J}_{a,\lambda}$ at the level $c_{a,\lambda}$. Since $\mathcal{J}_{a,\lambda}(u)=c_{a,\lambda}>0=\mathcal{J}_{a,\lambda}(0)$, we conclude that $u\not\equiv 0$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] Now to conclude the proof of Theorem \ref{thm1} we claim that \begin{equation}\label{claim2} \text{there exists $\lambda^*\geq\lambda_0$ such that $\|u_\lambda\|_{X_0}\leq t_0$ for all $\lambda\geq\lambda^*$}\,, \end{equation} where $t_0$ is defined in \eqref{trun}. Suppose not, then there exists a sequence $\{\lambda_j\}$ in $\mathbb{R}$ such that $\|u_{\lambda_j}\|_{X_0}\geq t_0$. Since $u_{\lambda_j}$ is a critical point of the functional $\mathcal{J}_{a,\lambda_j}$ which implies \begin{align*} c_{a,\lambda_j} &\geq\frac{1}{p}\widehat{M_a}(\|u_{\lambda_j}\|_{X_0}^{p}) -\frac{1}{\sigma}M_a(\|u_{\lambda_j}\|_{X_0}^{p}) \|u_{\lambda_j}\|_{X_0}^{p}\\ &\geq\big(\frac{1}{p}m_0-\frac{1}{\sigma}a\big)\|u_{\lambda_j}\|_{X_0}^{p}\\ &\geq\big(\frac{1}{p}m_0-\frac{1}{\sigma}a\big)t^{p}_{0}, \end{align*} which contradicts Lemma \ref{infinito}, since $m_0< a< \frac{\sigma}{p}m_0$. Hence $u_\lambda$ is a solution of problem $(M_\lambda)$. Since $c_{a,\lambda}\to 0$ as $\lambda\to 0$, implies $ \|u_\lambda\|_{X_0}\to0$ as $\lambda\to\infty$. \smallskip Now, we claim that $u_{\lambda}$ is non-negative in $\mathbb{R}^{n}$. Take $v=u^{-}\in X_0$, in \eqref{derivata}, where $u^{-}=\max(-u,0)$. Then \begin{align*} &M(\|u_\lambda\|_{X_0}^p)\int_{Q}|u_\lambda(x)-u_\lambda(y)|^{p-2} (u_\lambda(x)-u_\lambda(y))(u_\lambda^{-}(x)-u_\lambda^{-}(y))K(x-y) dxdy \\ &=\int_{\Omega} f(x,u_\lambda) u_\lambda^{-}(x) dx +\int_\Omega|u^-_\lambda(x)|^{p^*}dx. \end{align*} Now, using \[ (u_\lambda(x)-u_\lambda(y))(u_\lambda^{-}(x)-u_\lambda^{-}(y)) \leq -|u_\lambda^{-}(x)-u_\lambda^{-}(y)|^2 \] and $f(x,u_\lambda(x))u^-_\lambda(x)=0$ for a.e. $x\in \mathbb{R}^n$ we obtain \[ 0\leq -M(\|u_\lambda\|_{X_0}^p)\int_{Q}|u^-_\lambda(x)-u^-_\lambda(y)|^{p}K(x-y) -\int_\Omega|u^-_\lambda(x)|^{p^*}dx\leq-m_0\|u^-_\lambda\|^p_{X_0}. \] Thus $\|u_\lambda^{-}\|_{X_0}=0$ and hence $u_\lambda>0$. \end{proof} \begin{thebibliography}{00} \bibitem{acf} C. O. Alves, F. J. S. A. Correa, G. M. Figueiredo; \emph{On a class of nonlocal elliptic problems with critical growth,} Differ. Equ. Appl., 2, (2010), 409--417. \bibitem{app} D. 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