\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 321, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/321\hfil Fractional Schr\"odinger equations] {Existence of high-energy solutions for supercritical fractional Schr\"odinger \\ equations in $\mathbb{R}^N$} \author[L. Gan, W. Liu \hfil EJDE-2016/321\hfilneg] {Lu Gan, Weiming Liu} \address{Lu Gan \newline School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China} \email{ganlu0401@163.com} \address{Weiming Liu (corresponding author)\newline School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China} \email{whu.027@163.com} \thanks{Submitted October 31, 2016. Published December 20, 2016.} \subjclass[2010]{35J20, 35B09, 35J60} \keywords{Fractional Schr\"odinger equation; supercritical; \hfill\break\indent finite-dimensional reduction method} \begin{abstract} In this article, we study supercritical fractional Schr\"odinger equations. Applying the finite-dimensional reduction method and the penalization method, we obtain the high-energy solutions for this equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction and statement of main results}\label{s1} This article is devoted to the study of the problem \begin{equation}\label{1.1} \begin{gathered} (-\Delta)^su=V(x)u^{2^{*}(s)+\epsilon-1},\quad u>0,\; x\in\mathbb{R}^N, \\ u\to 0\quad \text{as } |x|\to +\infty, \end{gathered} \end{equation} where $2^{*}(s)=\frac{2N}{N-2s}$, $N>2s$, $00$, $V$ is a positive continuous potential. Here, the fractional Laplacian of a function $f:\mathbb{R}^N \to \mathbb{R}$ is expressed by the formula \begin{equation}\label{1.2} \begin{aligned} (-\Delta\big)^sf(x) &=C_{N,s}\operatorname{p.v.}\int_{\mathbb{R}^N}\frac{f(x)-f(y)}{|x-y|^{N+2s}}dy\\ &=C_{N,s}\lim_{\delta\to 0}\int_{\mathbb{R}^N\setminus B_{\delta}(x)} \frac{f(x)-f(y)}{|x-y|^{N+2s}}dy, \end{aligned} \end{equation} where $C_{N,s}$ is some normalization constant. The operator $(-\Delta)^s$ can be seen as the infinitesimal generators of L\'{e}vy stable diffusion processes (see \cite{a}). The L\'{e}vy processes occur widely in physics, biology, chemistry and finance (see \cite{a,a1}). The stable L\'{e}vy processes that give rise to equations with fractional Laplacians have recently attracted much research interest, and there are a lot of results in the literature on the existence of such solutions. In \cite{bc}, Barrios et al.\ studied the existence and multiplicity of solutions to the following critical problem with convex-concave nonlinearities \begin{equation}\label{1.333} \begin{gathered} (-\Delta)^su=\lambda u^{q}+u^{2^{*}(s)-1},\quad u>0,\; x\in\Omega, \\ u=0,\quad x\in\mathbb{R}^{N}\setminus\Omega. \end{gathered} \end{equation} As we know, the fractional power of the Laplacian can also be defined by using spectral decomposition. The same problem considered in \cite{bc} but for this spectral fractional Laplacian has been treated in \cite{bcd}. As in \cite{bcd} the purpose of this paper is to study the existence of weak solutions for \eqref{1.333}. In \cite{dm}, in order to construct solutions to the problem of the form $(-\Delta)^su=\epsilon h u_{+}^{q}+u_{+}^{p}$, Dipierro et al. used the Lyapunov-Schmidt reduction, that takes advantage of the variational structure of the problem. For related results, we refer the reader to \cite{cr,dds,dpv,jl1,jl2,lw,lg,sv,sv1,t}. Let us come back to equation \eqref{1.1}. Recently, many results on the existence of solutions for problem \eqref{1.1} when $\epsilon=0$ have been obtained. Liu \cite{l1} obtained infinitely many concentration solutions for \eqref{1.1} under certain conditions. Assume $V=1+\tau K$ and $K$ has at least two critical points satisfying some local conditions, Chen and Zheng \cite{cz} proved the existence of two-peak solutions when the positive number $\tau$ is small enough. When $V\equiv1$, the existence of finite-energy sign-changing solutions to \eqref{1.1} has been established by Garrido and Musso \cite{gm}. In particular, DelaTorre et al.\ \cite{dpg} constructed a class of Delaunay-type solutions for \eqref{1.1}. When $s=1$, problem \eqref{1.1} reduces formally, to the classical Schr\"odinger equation \begin{equation}\label{1.3} \begin{gathered} -\Delta u=V(x)u^{2^{*}+\epsilon-1},\quad u>0, \\ u\to 0\quad \text{as } |x|\to +\infty. \end{gathered} \end{equation} The study of problem \eqref{1.3} has attracted considerable attention in recent years, and there are several results in the literature on the existence of solutions. When $V$ is a perturbation of the constant, Ambrosetti et al.\ \cite{agp} and Cao et al.\ \cite{cn} proved the existence of two or many positive solutions. Li \cite{l1} proved that \eqref{1.3} has infinitely many positive solutions if $V$ is periodic, while similar result was obtained in \cite{y1} if $V$ has a sequence of strict local maximum points tending to infinity. Wei and Yan \cite{wy} obtained solutions with large number of bumps near infinity for \eqref{1.3} with $V$ being radial. Meanwhile, they proved that the energy of these solutions can be arbitrarily large. For related results, we refer the readers to \cite{cp,cpy,h,jl} and the references therein. The aim of this article is to show the existence of high-energy solutions for the fractional Schr\"odinger equation with slightly supercritical exponent. We assume that the positive continuous potential $V$ satisfies the following conditions: \begin{itemize} \item[(A1)] There exist constants $q\in [0, 2s)$ and $C>0$, such that $V(x)\leq C(1+|x|)^q$~for all $x\in \mathbb{R}^N$; \item[(A2)] For some $\mu, r>0$, $V\in C^{2, \mu}(B_r(y_0))$, and $\Delta V(y_0)>0, $ where $y_0\in \mathbb{R}^N$ is a strict local minimum point of $V$. \end{itemize} Now, we recall the basic theory on fractional Laplacian operator. For $s\in (0, 1)$, the nonlocal operator $(-\Delta)^s$ in $\mathbb{R}^{N}$ is defined on the Schwartz class through the Fourier transform $$ \widehat{(-\Delta)^s}f(\xi) = |\xi|^{2s}\widehat{f}(\xi), $$ or via the Riesz potential. And $\widehat{}$\, is the Fourier transform. When $f$ has some sufficiently regularity, the fractional Laplacian of a function $f: \mathbb{R}^{N} \to \mathbb{R}$ is expressed as \eqref{1.2}. That integral makes sense directly when $s < 1/2$ and $f \in C^{0,\gamma}(\mathbb{R}^{N})$ with $\gamma > 2s$, or if $f \in C^{1,\gamma}(\mathbb{R}^{N})$ with $1 + 2\gamma> 2s$. It is well known that $(-\Delta)^s$ on $\mathbb{R}^{N}$ with $0 < s < 1$ is a nonlocal operator. In the remarkable work by Caffarelli and Silvestre \cite{cs}, this nonlocal operator was expressed as a generalized Dirichlet-to-Neumann map for a certain elliptic boundary value problem with a local differential operator defined on the upper half-space $\mathbb{R}^{N+1}_{+}:= \{(x, y) : x \in \mathbb{R}^{N}, y >0\}$. That is, for a function $f: \mathbb{R}^{N} \to \mathbb{R}$, we consider the extension $u:\mathbb{R}^{N} \times [0,+\infty) \to \mathbb{R}$ that satisfies the equations \begin{gather}\label{2.2} u(x, 0) = f(x), \\ \label{2.3} \Delta_{x}u +\frac{1 - 2s}{y}u_{y} + u_{yy} = 0. \end{gather} Equation \eqref{2.3} can also be written as \begin{equation}\label{2.4} \operatorname{div}(y^{1-2s}\nabla u) = 0, \end{equation} which is clearly the Euler-Lagrange equation for the functional $$ J(u) =\int_{y>0}|\nabla u|^2y^{1-2s}\,dx\,dy. $$ From \eqref{2.2}-\eqref{2.4} it follows that $$ C(-\Delta)^sf = \lim_{y\to 0^{+}}-y^{1-2s}u_{y} =\frac{1}{2s}\lim_{y\to 0^{+}}\frac{u(x, y)-u(x, 0)}{y^{2s}}. $$ In the rest of this article, the homogeneous fractional Sobolev space is given by \[ D^s(\mathbb{R}^N)=\Bigl\{u\in L^{\frac{2N}{N-2s}}(\mathbb{R}^N): \int_{\mathbb{R}^N}|\xi|^{2s}|\widehat{u}(\xi)|^2<+\infty\Bigl\} \] with the norm $$ \|u\|=\Bigl(\int_{\mathbb{R}^N}|\xi|^{2s}|\widehat{u}(\xi)|^2\Bigl)^{1/2}, $$ which is induced by the inner product \[ \langle u,v \rangle_{s} =\Bigl(\int_{\mathbb{R}^N}|\xi|^{2s}\widehat{u}(\xi)\widehat{v}(\xi)\Bigl)^{1/2}. \] The so-called Gagliardo semi-norm of $u$ is defined as $$ [u]_{H^s(\mathbb{R}^{N})} :=\Bigl(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dx\,dy\Bigl)^{1/2}. $$ It can be proved \cite[Proposition 3.4 and 3.6]{npv} that $$ [u]_{H^s(\mathbb{R}^{N})} =C\Bigl(\int_{\mathbb{R}^N}|\xi|^{2s}|\widehat{u}(\xi)|^2\Bigl)^{1/2} =C\|(-\Delta)^{\frac{s}{2}}u\|_{L^2(\mathbb{R}^{N})} $$ for a suitable positive constant $C$ depending only on $s$ and $N$. We consider the equation \begin{equation}\label{1.4} (-\Delta)^su=u^{2^{*}(s)-1},~u>0~\text{on}~\mathbb{R}^N. \end{equation} It has been proved in \cite{clo,l} that the following function, for $y\in\mathbb{R}^{N}$ and $\lambda>0$, $$ U_{y,\lambda}=C_0\Bigl(\frac{\lambda}{1+\lambda^2|x-y|^2}\Bigl) ^{\frac{N-2s}{2}},\quad x\in\mathbb{R}^{N}, $$ where $C_0=C_0(N,s)>0$, solves \eqref{1.4} on $\mathbb{R}^{N}$. For any positive integer $m$, $\mathbf{y}=(y_{1},y_{2},\ldots,y_{m})\in\mathbb{R}^{mN}$, $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{m})$, and $\lambda_{k}>0$, $k=1,2,\ldots,m$, we define \begin{align*} E_{\mathbf{y},\lambda} =\Bigl\{&w\in D^s(\mathbb{R}^{N}): \big\langle w, \frac{\partial U_{y_{k},\lambda_{k}}} {\partial \lambda_{k}}\big\rangle_{s} =\big\langle w,\frac{\partial U_{y_{k},\lambda_{k}}} {\partial y_{ki}} \big\rangle_{s}=0,\\ & k=1,\ldots, m,\; i=1,\ldots, N \Bigl\}. \end{align*} Our main result in this paper can be stated as follows. \begin{theorem}\label{thm1.1} Suppose that $N>2s$, $00$, such that for each $\varepsilon\in (0, \varepsilon_0]$, the problem \eqref{1.1} has a solution of the form $u_\varepsilon=\sum_{k=1}^{m}U_{y_{\varepsilon_k},\lambda_{\varepsilon_k}} +\omega_\varepsilon$, where $\omega_\varepsilon\in E_{\mathbf{y},\lambda}$, $\mathbf{y}=(y_{\epsilon_{1}},y_{\epsilon_{2}},\ldots,y_{\epsilon_{m}})$ and as $\varepsilon\to 0$, $y_{\varepsilon_k}\to y_0$, $\lambda_{\varepsilon_k}\to +\infty$, $k=1, \ldots, m$, $\|\omega_\varepsilon\|\to 0$, \end{theorem} The proof of our results is inspired by the methods of \cite{dpp,y}, we will combine a penalization argument and Lyapunov-Schmidt reduction scheme which are similar to \cite{dpp,y} to prove our main result. This article is organized as follows. In section \ref{s2}, we introduce the penalization problem, give some preliminary estimates and carry out the finite dimensional reduction. In section \ref{s3}, we give the proof of Theorem \ref{thm1.1}. Some technical estimates are left in the appendix. Throughout this paper, we simply write $\int f$ to mean the Lebesgue integral of $f(x)$ in $\mathbb{R}^{N}$. The ordinary inner product between two vectors $a,b\in \mathbb{R}^{N}$ will be denoted by $a\cdot b$, and $C, \tilde{C},c_{i}$ denote generic constants, which may vary inside a chain of inequalities. We use $O(t),o(t)$ to mean $|O(t)|\leq C|t|,\frac{o(t)}{t}\to 0$ as $t\to 0$; $o(1)$ denotes quantities that tend to 0 as $|x|\to \infty$. % page 4 \section{Preliminaries and finite dimensional reduction}\label{s2} In this section, we give some preliminary results, which are crucial in the proof of the main theorem and the finite-dimensional reduction. Problem \eqref{1.1} is the Euler-Lagrange equation of functional $$ I_{1}(u)=\frac{1}{2}\langle u, u\rangle_s -\frac{1}{2^{*}(s)+\epsilon}\int_{\mathbb{R}^N}V(x)u^{2^{*}(s)+\epsilon}. $$ As we know, under the conditions (A1) and (A2), the functional $I_{1}(u)$ will not be well defined and differentiable in $D^s(\mathbb{R}^{N})$ Inspired by the idea introduced by Yan \cite{y} and Deng et al.\ \cite{dpp}, we modify the nonlinearity as in \cite{y}. To this end, we need to fix some notation. Choose $R>0$ large enough. Define $$ f(x,u)=\chi_{B_{R}(0)}(x)f_{1}(u)+\chi_{B^{C}_{R}(0)}(x)f_{2}(u), $$ where $\chi_{B_{R}(0)}$ denotes the characteristic function of $B_{R}(0)$, and \[ f_{1}(u)= \begin{cases} u^{2^{*}(s)+\epsilon-1},& \text{if } 0\leq u\leq\epsilon^{-k_{2}N}, \\ a_{\epsilon}u^{2^{*}(s)-1}+b_{\epsilon},& \text{if } u\geq\epsilon^{-k_{2}N}, \\ -f_{1}(-u),& \text{if } u<0, \end{cases} \] where $$ a_{\epsilon}=\Bigl(1+\frac{\epsilon}{2^{*}(s)-1}\Bigl)\epsilon^{-k_{2}N\epsilon},\quad b_{\epsilon}=-\frac{\epsilon}{2^{*}(s)-1}\epsilon^{-k_{2}N(2^{*}(s)+\epsilon-1)}, $$ and $k_{2}>0$ is a constant to be determined in Proposition \ref{prop3.1}. Moreover, $$ f_{2}(x,u)=\frac{1}{|x|^{N+2s+\epsilon(N-2s)}}\bar{f}_{2}(|x|^{N-2s}u), $$ where the nonnegative $C^{1}$ function $\bar{f}_{2}$ satisfies: \[ \bar{f}_{2}(u)=\begin{cases} 0 &\text{if }u\geq2,\\ u^{2^{*}(s)+\epsilon-1} &\text{if }u\in[0,1], \\ -\bar{f}_{2}(-u) &\text{if }u<0. \end{cases} \] Let $F(x,u)=\int_0^{u}f(x,\tau)d\tau$, then we have the following Lemma. \begin{lemma}\label{lem2.1} Assume that $V(x)$ satisfies (A1). Then $\int_{\mathbb{R}^{N}}V(x)F(x,u)$ is well defined on $D^s(\mathbb{R}^{N})$. \end{lemma} \begin{proof} Using (A1) and the definition of $f(x,u)$, we obtain \begin{align*} &\bigl|\int_{\mathbb{R}^N}V(x)F(x, u)dx\bigl| \\ &\leq \int_{\mathbb{R}^N\setminus B_R(0)}|V(x)F(x, u)|dx +\int_{B_R(0)}|V(x)F(x, u)|dx\\ &\leq C\int_{\mathbb{R}^N\setminus B_R(0)}(1+|x|)^q \Bigl(\int_0^u f(x, \tau)d\tau\Bigl)dx+C\int_{B_R(0)}|u|^{2^*(s)}dx\\ &\leq C\int_{\mathbb{R}^N\setminus B_R(0)}(1+|x|)^q \frac{1}{|x|^{N+2s+\varepsilon(N-2s)}} \Bigl(\int_0^u \bar{f_2}(|x|^{N-2s}\tau)d\tau\Bigl)dx \\ &\quad +C\int_{\mathbb{R}^N}|u|^{2^*(s)}dx\\ &\leq C\int_{\mathbb{R}^N\setminus B_R(0)}(1+|x|)^q \frac{1}{|x|^{2N+\varepsilon(N-2s)}}\Bigl(\int_0^{|x|^{N-2s}u} \bar{f_2}(\tau)d\tau\Bigl)dx +C\int_{\mathbb{R}^N}|u|^{2^*(s)}dx\\ &\leq C\int_{\mathbb{R}^N}\frac{1}{1+|x|^{2N-q+(N-2s)\varepsilon}}dx+ C\int_{\mathbb{R}^N}|u|^{2^*(s)}dx<+\infty. \end{align*} Consequently, the result follows from the above estimate. \end{proof} Now we consider the penalization problem \begin{equation}\label{2.5} \begin{gathered} (-\Delta)^s u=V(x)f(x, u),\quad x\in {R}^{N}, \\ u\in D^s(\mathbb{R}^N). \end{gathered} \end{equation} The functional associated with problem \eqref{2.5} is given by \begin{equation}\label{2.6} I(u)=\frac{1}{2}\langle u, u\rangle_s-\int_{\mathbb{R}^N}V(x)F(x, u), \quad u\in D^s(\mathbb{R}^N). \end{equation} It follows from Lemma \ref{lem2.1} that $I\in C^{1}$ is well defined in $E_{\mathbf{y},\lambda}$ and hence its critical points are solutions of problem \eqref{2.5}. Denote $$ U(\mathbf{y}, \mathbf{\lambda})=\sum_{k=1}^{m}U_{y_k, \lambda_k} $$ and set \begin{equation}\label{2.7} J(\mathbf{y}, \mathbf{\lambda}, w)=I\Bigl(\sum_{k=1}^{m}U_{y_k, \lambda_k}+w\Bigl), \quad \forall (\mathbf{y}, \mathbf{\lambda}, w)\in M_{\mathbf{y}, \mathbf{\lambda}}, \end{equation} where \begin{equation}\label{2.8} M_{\mathbf{y}, \mathbf{\lambda}}=\{(\mathbf{y}, \mathbf{\lambda}, w): w\in E_{\mathbf{y},\lambda}, (\mathbf{y}, \mathbf{\lambda})\in D_{\mathbf{y}, m}, \|w\|\leq \delta\}, \end{equation} \begin{align*} D_{\mathbf{y}, m}= \Bigl\{&\mathbf{y}=(y_1, y_2, \ldots, y_m)\in \mathbb{R}^{mN}, \mathbf{\lambda}=(\lambda_1, \lambda_2, \ldots, \lambda_m), y_k\in \overline{B_\delta(y_0)}, \\ &\lambda_k\in [\epsilon^{-k_1},\epsilon^{-k_2}],\; k=1, \ldots,m,\; \epsilon_{jk}\leq\frac{1}{L},\; j\neq k\Bigl\}, \end{align*} where small $k_1>0$ and large $k_2>0$ are constants to be determined in Proposition \ref{prop3.1}, $L>0$ large enough, and \begin{align*} \epsilon_{jk}=\Bigl(\frac{\lambda_{j}}{\lambda_{k}} +\frac{\lambda_{k}}{\lambda_{j}}+\lambda_{j}\lambda_{k}|y_{j}-y_{k}|^2 \Bigl)^{\frac{2s-N}{2}}. \end{align*} \begin{lemma}\label{lem2.2} There exists $\varepsilon_0>0$, such that, for $\varepsilon\in (0, \varepsilon_0]$, $L>0$ large enough, $\delta>0$ small enough, $(\mathbf{y}, \lambda, w)\in M_{\mathbf{y}, \mathbf{\lambda}}$ is a critical point of $J$ if and only if $u=\sum_{k=1}^{m}U_{y_k, \lambda_k}+w$ is a critical point of $I$ in $D^s(\mathbb{R}^N)$. \end{lemma} The proof of Lemma \ref{lem2.2} is standard, since can be complete it with the same arguments as those in \cite{ly,cz}, we omit it. Without loss of generality, we assume that $y_0=0$ and $V(0)=1$. Expanding $J(\mathbf{y}, \lambda, w)$, we obtain $$ J(\mathbf{y}, \mathbf{\lambda}, w)=J(\mathbf{y}, \mathbf{\lambda}, 0)+l_{\mathbf{y}, \mathbf{\lambda}}(w)+\frac{1}{2}\langle L_{\mathbf{y}, \mathbf{\lambda}}w, w\rangle_{s}+R_{\mathbf{y}, \mathbf{\lambda}}(w), $$ where \begin{gather*} l_{\mathbf{y}, \mathbf{\lambda}}(w) =-\int_{\mathbb{R}^N}V(x)\Bigl(\sum_{k=1}^{m}U_{y_k, \lambda_k} \Bigl)^{2^*(s)+\varepsilon-1}w+\langle U_{\mathbf{y}, \mathbf{\lambda}}, w\rangle_s, \\ \langle L_{\mathbf{y}, \mathbf{\lambda}}w, w\rangle_{s} =\langle w, w\rangle_s-(2^*(s)+\varepsilon-1)\int_{\mathbb{R}^N}V(x) \Bigl(\sum_{k=1}^{m}U_{y_k, \lambda_k}\Bigl)^{2^*(s)+\varepsilon-2}w^2, \\ \begin{aligned} R_{\mathbf{y}, \mathbf{\lambda}, w} &=-\int_{\mathbb{R}^N}V(x)F(x, U_{\mathbf{y}, \mathbf{\lambda}}+w) +\int_{\mathbb{R}^N}V(x)F(x, U_{\mathbf{y}, \mathbf{\lambda}}) \\ &\quad +\int_{\mathbb{R}^N}V(x)\Bigl(\sum_{k=1}^{m}U_{y_k, \lambda_k}\Bigl)^{2^*(s) +\varepsilon-1}w \\ &\quad +\frac{2^*(s)+\varepsilon-1}{2}\int_{\mathbb{R}^N}V(x) \Bigl(\sum_{k=1}^{m}U_{y_k, \lambda_k}\Bigl)^{2^*(s)+\varepsilon-2}w^2. \end{aligned} \end{gather*} Now, we state a lemma which is very important for our precise estimate on the functional energy and can be found in \cite{ny}. \begin{lemma}\label{lem2.a} For $2|b|$, $$ \Bigl||a+b|^{q}-|a|^{q}-|b|^{q}-q|a|^{q-1}|b|-q|b|^{q-2}|a|\Bigl| \leq C|b|^{q-1}|a|. $$ For $q>3$, $$ \Bigl||a+b|^{q}-|a|^{q}-|b|^{q}-q|a|^{q-1}|b|-q|b|^{q-1}|a|\Bigl| \leq C(|a|^{q-2}|b|^2+|b|^{q-2}|a|^2). $$ \end{lemma} Next, we show the invertibility of $L_{\mathbf{y}, \mathbf{\lambda}}$. \begin{lemma}\label{lem2.3} There exist constants $\varepsilon_0>0$ and $C>0$ such that for $(\mathbf{y}, \mathbf{\lambda})\in D_{\mathbf{y}, m}$, $$ \|L_{\mathbf{y}, \mathbf{\lambda}}w\|\geq C\|w\|, \quad \forall w\in E_{\mathbf{y},\lambda}. $$ \end{lemma} \begin{proof} We proceed by contradiction. Assume that there exist $\varepsilon_n\to 0, \delta_n\to 0, \mathbf{\lambda}^n\to \infty$, $(\mathbf{y}^n, \lambda^n)=(y_{1}^n, \ldots, y_{m}^n$, $\lambda_{1}^n, \ldots, \lambda_{m}^n)\in D_{\mathbf{y}, m}$ and $w_{n}\in E_{\mathbf{y},\lambda}$, such that \begin{equation}\label{2.9} \langle L_{\mathbf{y}^n, \lambda^n}w_n, \varphi\rangle_{s}=o_n(1)\|w_n\|\|\varphi\|, \quad \forall\varphi\in E_{\mathbf{y},\lambda}. \end{equation} Without loss of generality, we assume that $\|w_n\|=1$. Let \[ \tilde{w}_{n, k}(x)=(\lambda_{k}^n)^{\frac{2s-N}{2}} w_n((\lambda_{k}^n)^{-1}x+y_{k}^n), \quad k=1, \ldots, m. \] Assume that \begin{gather*} \tilde{w}_{n, k}\rightharpoonup w_k^*, \quad k=1, \ldots, m, \quad\text{as } n\to \infty, \\ \tilde{w}_{n, k}\to w_k^*, \quad\text{strongly in } L_{loc}^2(\mathbb{R}^N), \quad k=1, \ldots, m, \quad\text{as }~n\to \infty. \end{gather*} From $$ \Bigl\langle\frac{\partial U_{y_k^n, \lambda_k^n}}{\partial \lambda_{k}^n}, w_n\Bigl\rangle_{s}=\Bigl\langle\frac{\partial U_{y_k^n, \lambda_k^n}}{\partial y_{k_i}^n}, w_n\Bigl\rangle_{s}=0, $$ $k=1, \ldots , m, i=1, \ldots,N$, we obtain $$ \Bigl\langle\frac{\partial U_{0, 1}}{\partial \lambda}\Bigl|_{\lambda=1}, \tilde{w}_{n, k}\Bigl\rangle_{s}=\Bigl\langle\frac{\partial U_{0, 1}}{\partial x_{i}} \Bigl|_{x=0}, \tilde{w}_{n, k}\Bigl\rangle_{s}=0, $$ for $k=1, \ldots, m$, $i=1, \ldots, N$. So $w_k^*$ satisfies \begin{equation}\label{2.10} \Bigl\langle\frac{\partial U_{0, 1}}{\partial \lambda}\Bigl|_{\lambda=1}, w_k^*\Bigl\rangle_{s}=\Bigl\langle\frac{\partial U_{0, 1}}{\partial x_{i}} \Bigl|_{x=0}, w_k^*\Bigl\rangle_{s}=0, \end{equation} for $k=1, \ldots, m, i=1, \ldots, N$. Define $$ \widetilde{E}_{\mathbf{y},\lambda} =\{\varphi\in D^s(\mathbb{R}^N), \Bigl\langle \frac{\partial U_{0, 1}}{\partial \lambda}\Bigl|_{\lambda=1}, \varphi\Bigl\rangle_{s} =\Bigl\langle\frac{\partial U_{0, 1}}{\partial x_i}\Bigl|_{x=0}, \varphi\Bigl\rangle_{s}=0, i=1, \ldots, N\}. $$ Note that \begin{equation}\label{2.11} o(1)\|\varphi\|=\langle w_n, \varphi\rangle_s -\int_{\mathbb{R}^N}V(x)f'(x, U_{\mathbf{y}^n, \lambda^n})w_{n}\varphi. \end{equation} Let $\varphi\in C_0^\infty(\mathbb{R}^N)\cap \widetilde{E}_{\mathbf{y},\lambda}$ and take $\varphi_n(x):=(\lambda_k^n)^{\frac{2s-N}{2}}\varphi[(x-y_k^n)\lambda_k^n]$. Letting $n\to \infty$, we obtain $$ \langle w_k^*, \varphi\rangle_s-(2^*(s)-1) \int_{\mathbb{R}^N}U_{0, 1}^{2^*(s)-2}w_{k}^{*}\varphi=0. $$ It is easy to prove that \begin{equation}\label{2.12} \langle w_k^*, \varphi\rangle_s-(2^*(s)-1) \int_{\mathbb{R}^N}U_{0, 1}^{2^*(s)-2}w_{k}^{*}\varphi=0,\quad \forall\varphi\in \widetilde{E}_{\mathbf{y},\lambda}. \end{equation} But \eqref{2.12} is true for $\varphi= c_0\frac{\partial U_{0, 1}}{\partial \lambda}\bigl|_{\lambda=1} +\sum_{i=1}^{n}c_{i}\frac{\partial U_{0, 1}}{\partial x_{i}}\bigl|_{x=0}$. Thus, \eqref{2.12} is true for any $\varphi\in E_{\mathbf{y},\lambda}$, and hence $w_k^*=c_0\frac{\partial U_{0, 1}}{\partial \lambda}\Bigl|_{\lambda=1} +\sum_{i=1}^{n}c_{i}\frac{\partial U_{0, 1}}{\partial x_{i}}\Bigl|_{x=0}$. It follows from \eqref{2.10} that $c_i=0~(i=0,1,\ldots,N)$ and $w_k^*=0$. Therefore, letting $\varepsilon_n, \delta_n>0$ small enough, $\lambda^n$ big enough, \begin{equation}\label{2.13} \begin{aligned} o(1)&=\langle w_n, w_n\rangle_s-\int_{\mathbb{R}^N}V(x)f'(x, U_{\mathbf{y}^{n}, \lambda^{n}})w_n^2 \\ & \geq 1-C\int_{\mathbb{R}^N}U_{0, 1}^{2^*(s)-2}w_n^2=1+O_R(1)+o(1). \end{aligned} \end{equation} This contradicts \eqref{2.9}. \end{proof} \begin{proposition}\label{prop2.4} For $\varepsilon>0$ sufficiently small and $(\mathbf{y}, \lambda)\in D_{\mathbf{y},m}$, there exists a $C^{1}$-map $w(\mathbf{y}, \lambda):D_{\mathbf{y},m}\to M_{\mathbf{y},\lambda}$ such that $w(\mathbf{y}, \lambda)$ satisfies $\Bigl\langle\frac{\partial J(w)}{\partial w}, \varphi\Bigl\rangle =0$ for all $\varphi\in M_{\mathbf{y},\lambda}$. Moreover, \begin{equation}\label{2.14} \|w\|\leq C\Bigl(\sum_{k=1}^{m}\Bigl(\frac{|DV(y_k)|}{\lambda_k} +\frac{1}{\lambda_k^2}+\frac{1}{\lambda_k^{\frac{N+2s}{2}}}|V(0)-V(y_k)| +\varepsilon \ln\lambda_k\Bigl)+\sum_{j\neq k}\varepsilon_{jk}^{\frac{1}{2} +\tau}\Bigl). \end{equation} \end{proposition} \begin{proof} To find a critical point for $J(w)$, we only need to solve \begin{equation}\label{2.15} l_{\mathbf{y}, \lambda}+\langle L_{\mathbf{y}, \lambda}w, w\rangle+\mathbb{R}^{'}(w)=0. \end{equation} From Lemma \ref{lem2.3}, we know that $L_{\mathbf{y}, \lambda}$ is invertible. Therefore, \eqref{2.15} can be rewritten as $$ w=\mathcal{A}(w)=:-L^{-1}_{\mathbf{y}, \lambda}l_{\mathbf{y}, \lambda}-L^{-1}_{\mathbf{y}, \lambda}\mathbb{R}^{'}(w).$$ Set \begin{align*} \mathcal{N}= \Bigl\{&w\in E_{\mathbf{y},\lambda}: \|w\|\leq \sum_{k=1}^{m}\frac{|DV(y_k)|}{\lambda_k^{1-\delta}} +\frac{1}{\lambda_k^{2(1-\delta)}} +\frac{1}{\lambda_k^{\frac{N+2s}{2}(1-\delta)}}|V(0)-V(y_k)|\\ &+\varepsilon^{1-\delta}\ln\lambda_k +\sum_{j\neq k}\varepsilon_{jk}^{(\frac{1}{2}+\tau)(1-\delta)}\Bigl\}, \end{align*} where $\delta>0$ is small enough. As in \cite{y}, $R(w)$ is the higher order term satisfying $$ \mathbb{R}^i(w)=O(\|w\|^{2+\theta-i}), \quad i=0, 1, 2, $$ where $\theta>0$ is some constant. Hence, Lemma \ref{lem2.5} below implies \begin{equation}\label{2.16} \begin{aligned} &\|\mathcal{A}(w)\| \\ &\leq C\|l_{y, \lambda}\|+C\|w\|^{1+\theta} \\ &\leq C\Bigl(\sum_{k=1}^{m}\Bigl(\frac{|DV(y_k)|}{\lambda_k} +\frac{1}{\lambda_k^2}+\frac{1}{\lambda_k^{\frac{N+2s}{2}}}|V(0)-V(y_k)| +\varepsilon \ln\lambda_k\Bigl)+\sum_{j\neq k}\varepsilon_{jk} ^{\frac{1}{2}+\tau}\Bigl) \\ &\quad +C \Bigl(\sum_{k=1}^{m}\frac{|DV(y_k)|}{\lambda_k^{1-\delta}} +\frac{1}{\lambda_k^{2(1-\delta)}} +\frac{1}{\lambda_k^{\frac{N+2s}{2}(1-\delta)}}|V(0)-V(y_k)| +\varepsilon^{1-\delta}\ln\lambda_k \\ & \quad +\sum_{j\neq k}\varepsilon_{jk}^{(\frac{1}{2}+\tau)(1-\delta)} \Bigl)^{1+\theta} \\ &\leq \sum_{k=1}^{m}\frac{|DV(y_k)|}{\lambda_k^{1-\delta}} +\frac{1}{\lambda_k^{2(1-\delta)}} +\frac{1}{\lambda_k^{\frac{N+2s}{2}(1-\delta)}}|V(0)-V(y_k)| +\varepsilon^{1-\delta}\ln\lambda_k \\ &\quad +\sum_{j\neq k}\varepsilon_{jk}^{(\frac{1}{2}+\tau)(1-\delta)}. \end{aligned} \end{equation} Meanwhile, \begin{align*} \|\mathcal{A}(w_1)-\mathcal{A}(w_2)\| &=\|L_{y, \lambda}^{-1}\mathbb{R}^{'}(w_1)-L_{y, \lambda}^{-1}\mathbb{R}^{'}(w_2)\| \\ & \leq C\|\mathbb{R}^{'}(w_1)-\mathbb{R}^{'}(w_2)\| \\ &\leq C\|\mathbb{R}^{''}(\varepsilon w_1+(1-\varepsilon)w_2)\|\|w_1-w_2\| \\ & \leq C(\|w_1\|^\theta+\|w_2\|^\theta)\|w_1-w_2\|\leq \frac{1}{2}\|w_1-w_2\|, \end{align*} where $\varepsilon \in (0, 1)$. Thus, $\mathcal{A}$ maps $\mathcal{N}$ to $\mathcal{N}$ and $\mathcal{A}$ is a contraction map. By the contraction mapping theorem, we see that there is a unique $w$ such that \eqref{2.15} holds, and from \eqref{2.16} that \eqref{2.14} holds. \end{proof} \begin{lemma}\label{lem2.5} $$ \|l_{\mathbf{y}, \lambda}\|\leq C\Bigl(\sum_{k=1}^{m} \Bigl(\frac{|DV(y_k)|}{\lambda_k}+\frac{1}{\lambda_k^2} +\frac{1}{\lambda_k^{\frac{N+2s}{2}}}|V(0)-V(y_k)| +\varepsilon \ln\lambda_k\Bigl)+\sum_{j\neq k}\varepsilon_{jk}^{\frac{1}{2} +\tau}\Bigl). $$ \end{lemma} \begin{proof} First, we know that \begin{equation}\label{2.17} \langle U_{\mathbf{y}, \lambda}, w\rangle_s =\Bigl\langle\sum_{k=1}^{m}U_{y_k, \lambda_k}, w\Bigl\rangle_s =\sum_{k=1}^{m}\int_{\mathbb{R}^N} U_{y_k, \lambda_k}^{\frac{N+2s}{N-2s}}w. \end{equation} Next, we estimate \begin{align*} &\int_{\mathbb{R}^N}V(x)f(x, U_{\mathbf{y}, \lambda})w \\ &=\int_{\mathbb{R}^N}V(x)\Bigl(\sum_{k=1}^{m}U_{y_k, \lambda_k}\Bigl)^{2^*(s) +\varepsilon-1}w \\ &= \begin{cases} \sum_{k=1}^{m}\int_{\mathbb{R}^N}V(x)U_{y_k, \lambda_k}^{2^*(s)+\varepsilon-1}w\\ +O\Bigl(\sum_{j\neq k}\int_{\mathbb{R}^N}V(x)U_{y_j, \lambda_j}^{\frac{2^*(s) +\varepsilon-1}{2}}U_{y_k, \lambda_k}^{\frac{2^*(s)+\varepsilon-1}{2}}w\Bigl), & \text{if } 1<2^*(s)+\varepsilon-1\leq 2, \\[4pt] \sum_{k=1}^{m}\int_{\mathbb{R}^N}V(x)U_{y_k, \lambda_k}^{2^*(s) +\varepsilon-1}w+O\Bigl(\sum_{j\neq k}\varepsilon_{jk}\Bigl)\|w\|, & \text{if } 2^*(s)+\varepsilon-1>2, \end{cases} \\ &=\begin{cases} \sum_{k=1}^{m}\int_{\mathbb{R}^N}V(x)U_{y_k, \lambda_k}^{2^*(s) +\varepsilon-1}w+O\Bigl(\sum_{j\neq k}\varepsilon_{jk}^{\frac{1}{2}+\tau}\Bigl)\|w\|, & \text{if } 1<2^*(s)+\varepsilon-1\leq 2, \\ \sum_{k=1}^{m}\int_{\mathbb{R}^N}V(x)U_{y_k, \lambda_k}^{2^*(s)+\varepsilon-1}w +O\Bigl(\sum_{j\neq k}\varepsilon_{jk}\Bigl)\|w\|, & \text{if } 2^*(s)+\varepsilon-1>2. \end{cases} \end{align*} We also have \begin{equation}\label{2.19} \begin{aligned} &\int_{\mathbb{R}^N}V(x)U_{y_k, \lambda_k}^{2^*(s)+\varepsilon-1}w \\ &=\int_{\mathbb{R}^N}V(x)U_{y_k, \lambda_k}^{2^*(s)-1}w +O(\varepsilon\ln\lambda_k)\|w\| \\ &=\int_{\mathbb{R}^N} V(y_k)U_{y_k, \lambda_k}^{2^*(s)-1}w +O\Bigl(\frac{|DV(y_k)|}{\lambda_k}+\frac{1}{\lambda_k^2} +\varepsilon\ln\lambda_k\Bigl)\|w\|. \end{aligned} \end{equation} Combining above estimates, we obtain \begin{align*} |l_{y, \lambda}w| &=\sum_{k=1}^{m}\int_{\mathbb{R}^N} U_{y_k, \lambda_k}^{\frac{N+2s}{N-2s}}w -\sum_{k=1}^{m}\int_{\mathbb{R}^N} V(y_k)U_{y_k, \lambda_k}^{\frac{N+2s}{N-2s}}w \\ &+O\Bigl(\sum_{k=1}^{m}\Bigl(\frac{|DV(y_k)|}{\lambda_k} +\frac{1}{\lambda_k^2}+\varepsilon \ln\lambda_k\Bigl) +\sum_{j\neq k}\varepsilon_{jk}^{\frac{1}{2} +\tau}\Bigl)\|w\|\\ &=\sum_{k=1}^{m}\int_{\mathbb{R}^N} |V(0)-V(y_k)|U_{y_k,\lambda_k }^{\frac{N+2s}{N-2s}}w \\ &\quad +O\Bigl(\sum_{k=1}^{m}\Bigl(\frac{|DV(y_k)|}{\lambda_k} +\frac{1}{\lambda_k^2}+\varepsilon \ln\lambda_k\Bigl) +\sum_{j\neq k}\varepsilon_{jk}^{\frac{1}{2}+\tau}\Bigl)\|w\|\\ &=O\Bigl(\sum_{k=1}^{m}\Bigl(\frac{|DV(y_k)|}{\lambda_k} +\frac{1}{\lambda_k^2}+\frac{1}{\lambda_k^{\frac{N+2s}{2}}}|V(0)-V(y_k)| +\varepsilon \ln\lambda_k\Bigl) \\ &\quad +\sum_{j\neq k}\varepsilon_{jk}^{\frac{1}{2}+\tau}\Bigl)\|w\|. \end{align*} \end{proof} \section{Proof of main result}\label{s3} Let $ w(\mathbf{y}, \lambda)$ be the map obtained in Proposition \ref{prop2.4}. Define $$ \widetilde{I}(\mathbf{y}, \lambda):=I(\mathbf{y}, \lambda, w(\mathbf{y}, \lambda)), \quad (\mathbf{y}, \lambda)\in D_{\mathbf{y}, m}. $$ Let $(\mathbf{y}_\varepsilon, \lambda_\varepsilon)\in D_{\mathbf{y}, m}$ be any point for which \begin{equation}\label{3.1} \widetilde{I}(\mathbf{y}_\varepsilon, \lambda_\varepsilon) =\sup\{\widetilde{I}(\mathbf{y}, \lambda):(\mathbf{y}, \lambda)\in D_{\mathbf{y}, m}\}. \end{equation} The next Proposition shows that for small $\varepsilon>0$, $(\mathbf{y}_\varepsilon, \lambda_\varepsilon)$ is an interior point of $D_{\mathbf{y}, m}$, and hence a critical point of $\widetilde{I}$. \begin{proposition}\label{prop3.1} Let $(\mathbf{y}_\varepsilon, \lambda_\varepsilon)$ satisfy \eqref{3.1}. Then as $\varepsilon\to 0$, \begin{gather*} y_\varepsilon^k\to 0,\quad k=1,\ldots, m,\\ \lambda_\varepsilon^k\in [\epsilon^{-k_1},\epsilon^{-k_2}],\quad k=1,\ldots, m,~\text{for some positive constant }k_10$ small enough. If $\lambda_\varepsilon^k=\epsilon^{-k_2}$ for some $k$, then from \eqref{3.6}, we obtain \[ \frac{N-2s}{2}k_2\varepsilon \ln\frac{1}{\epsilon}V(y_{\epsilon}^{k}) \int_{\mathbb{R}^N} U^{2^*(s)}\leq (2^*(s) +\epsilon)mC\epsilon\ln\frac{1}{\epsilon}+O(\epsilon), \] which is impossible if we choose $k_2>\max\{\frac{2(2^*(s)+1)mC}{(N-2s)\int_{\mathbb{R}^N} U^{2^*(s)}},\frac{1}{2}\}$. Consequently, the result $\lambda_\varepsilon^k\in [\epsilon^{-k_1},\epsilon^{-k_2}]$ $(k=1,\ldots, m)$ follows from the above estimate. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] By Proposition \ref{prop3.1}, we can check that \eqref{3.1} is achieved by $(\mathbf{y}_\varepsilon, \lambda_\varepsilon)$ which is an interior point of $D_{\mathbf{y}, m}$ for small $\epsilon$, It follows from Proposition \ref{prop2.4} that $(\mathbf{y}_\varepsilon, \lambda_\varepsilon)$ is a critical point of $J$. Using Lemma \ref{lem2.2}, we can obtain $u=\sum_{k=1}^{m}U_{y_k, \lambda_k}+w$ is a critical point of $I$. Note that $$ (-\Delta)^s w=V(x)f\Bigl(x, \sum_{k=1}^{m}U_{y_k, \lambda_k}+w\Bigl) -\sum_{k=1}^{m}U_{y_k, \lambda_k}^{2^*(s)-1}. $$ On the other hand, $$ \Bigl|V(x)f\Bigl(x, \sum_{k=1}^{m}U_{y_k, \lambda_k}+w\Bigl) -\sum_{k=1}^{m}U_{y_k, \lambda_k}^{2^*(s)-1}\Bigl| \leq C|w|^{2^*(s)-1}+C\epsilon^{-\frac{k_{2}(N+2s)}{2}}. $$ Since $\|w_{\epsilon}\|\to 0$ as $\epsilon\to 0$, by adapting the same approach explored in \cite{am} and \cite{dp}, we deduce that $$ |w_{\epsilon}|\leq C\|w_{\epsilon}\|_{L^{2^*(s)}(B_{2}(0))} +C\epsilon^{-\frac{k_{2}(N+2s)}{2}},\quad \forall x \in B_{2}(0), $$ which implies $|u_{\epsilon}|\leq\epsilon^{-k_{2}N}$ for all $|x|0$ small enough, and it also follows from the same approach explored in \cite{am} and \cite{dp} that $$ |\bar{u}_{\epsilon}|_{L^{\infty}(B_{\gamma}(0))} \leq C\|\bar{u}_{\epsilon}\|_{L^{2^*(s)}(B_{2\gamma}(0))}\to 0, \quad \text{as }\epsilon\to 0, $$ which implies $|x|^{N-2s}u_{\epsilon}\to 0$ as $\epsilon\to 0$. From above estimates, we can obtain that $u$ solves indeed the original problem \eqref{1.1}. \end{proof} \section{Appendix}\label{s4} In this section, we prove some estimates needed in the proof of our main results. \begin{lemma}\label{lem4.1} For any $(\mathbf{y},\lambda)\in D_{\mathbf{y},m}$, we have \begin{align*} &I(U_{\mathbf{y}, \lambda})\\ &=\Bigl(\frac{m}{2}-\frac{1}{2^*(s)+\varepsilon} \sum_{k=1}^{m}V(y_k)\Bigl)\int_{\mathbb{R}^N} U^{2^*(s)} -\frac{1}{2^*(s)+\varepsilon}\sum_{k=1}^{m} \frac{\Delta V(y_k)}{2\lambda_k^2}\int_{\mathbb{R}^N} |x|^2U^{2^*(s)}\\ &\quad+O\Bigl(\sum_{k=1}^{m}\lambda_k^{-(2+\mu)}\Bigl) +O\Bigl(\sum_{k=1}^{m}\Bigl(\frac{\varepsilon \ln\lambda_k}{\lambda_k} +\varepsilon^2\ln^2\lambda_k\Bigl)\Bigl) +O\Bigl(\sum_{j\neq k}\Bigl(\varepsilon+\frac{1}{\lambda_j}\Bigl) \varepsilon_{jk}\Bigl)\\ & \quad-\frac{1}{2^*(s)+\varepsilon}\sum_{k=1}^{m}\Bigl(\varepsilon V(y_k) \int_{\mathbb{R}^N} \ln\lambda_k^{\frac{N-2s}{2}}U^{2^*(s)} -\varepsilon V(y_k)\int_{\mathbb{R}^N} U^{2^*(s)}\ln U\Bigl)\\ & \quad +O\Bigl(\sum_{k=1}^{m}(1-V(y_k))^{1+\tau}\Bigl) -\sum_{k=1}^{m-1}\int_{\mathbb{R}^N} V(x)U_{y_k, \lambda_k} \Bigl(\sum_{j=k+1}^{m}U_{y_j, \lambda_j}\Bigl)^{2^*(s)+\varepsilon-1} \\ &\quad +O\Bigl(\sum_{j\neq k}\varepsilon_{j, k}^{1+\tau}\Bigl). \end{align*} \end{lemma} \begin{proof} Firstly, for any $(\mathbf{y},\lambda)\in D_{\mathbf{y},m}$, we see that \begin{equation}\label{4.1} \begin{aligned} I(U_{\mathbf{y}, \lambda}) & =\frac{1}{2}\langle U_{\mathbf{y}, \lambda}, U_{\mathbf{y}, \lambda}\rangle_s-\int_{\mathbb{R}^N}V(x)F(x, U_{\mathbf{y}, \lambda})\\ &=\frac{1}{2}\langle U_{\mathbf{y}, \lambda}, U_{\mathbf{y}, \lambda}\rangle_s-\frac{1}{2^*(s)+\varepsilon}\int_{\mathbb{R}^N}V(x)U^{2^*(s) +\varepsilon}_{\mathbf{y}, \lambda}\\ &=\frac{1}{2}\sum_{k=1}^{m}\langle U_{y_k, \lambda_k}, U_{y_k, \lambda_k}\rangle_{s}+\sum_{j