\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 203, pp. 1--9.\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-2016/203\hfil Bifurcation and multiplicity of solutions] {Bifurcation and multiplicity of solutions for the fractional Laplacian with critical exponential nonlinearity} \author[P. K. Mishra, K. Sreenadh \hfil EJDE-2016/203\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 March 5, 2016. Published July 27, 2016.} \subjclass[2010]{35A15, 35R11} \keywords{Fractional Laplacian; bifurcation; exponential growth} \begin{abstract} We study the fractional elliptic equation \begin{gather*} (-\Delta)^{1/2} u = \lambda u+|u|^{p-2}ue^{u^2} ,\quad\text{in } (-1,1),\\ u=0\quad\text{in } \mathbb{R}\setminus(-1,1), \end{gather*} where $\lambda$ is a positive real parameter, $p>2$ and $(-\Delta)^{1/2}$ is the fractional Laplacian operator. We show the multiplicity of solutions for this problem using an abstract critical point theorem of literature in critical point theory. Precisely, we extended the result of Cerami, Fortuno and Struwe \cite{sfs} for the fractional Laplacian with exponential nonlinearity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction} We study the fractional elliptic equation \begin{equation} \label{ePl} \begin{gathered} (-\Delta)^{1/2} u = \lambda u+|u|^{p-2}ue^{u^2} ,\quad \text{in } (-1,1),\\ u=0\quad \text{in } \mathbb{R}\setminus(-1,1), \end{gathered} \end{equation} where $\lambda$ is a positive real parameter, $p>2$ and $(-\Delta)^{1/2}$ is the fractional Laplacian operator which is defined as follows \begin{equation*} -(-\Delta)^{1/2}u(x)={\frac{1}{2\pi}}\int_{\mathbb{R}} \frac{u(x+y)+u(x-y)-2u(x)}{|y|^{2}}dy\quad \text{for all } x\in \mathbb{R}. \end{equation*} The fractional Laplacian operator has been a classical topic in Fourier analysis and nonlinear partial differential equations for a long time. In the recent past the fractional Laplacian operator has been widely studied by many researchers due to its wide range of applications in many fields such as optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, soft thin films, semipermeable membranes, flame propagation, conservation laws, ultra-relativistic limits of quantum mechanics, quasi-geostrophic lows, multiple scattering, minimal surfaces, materials science, water waves, thin obstacle problem, optimal transport, image reconstruction and many more, see \cite{Levy, diff, ran, nondiff} and references therein. Bifurcation and multiplicity results for the case of Laplace operator with critical polynomial growth was initially studied in \cite{sfs}, where authors showed that the problem \begin{equation}\label{intr} -\Delta u=\lambda u+u^{2_*-2}u, \text{ in }\Omega,\quad u=0 \text{ on }\partial \Omega \end{equation} has $m$ pairs of solutions for $\lambda$ lying in the suitable left neighborhood of any eigenvalue with multiplicity $m$ of the Laplace operator with Dirichlet boundary conditions. Moreover the asymptotic behavior of the solutions is studied when the length of neighborhood goes to zero. Further the bifurcation and multiplicity results for the quasilinear counterpart of the problem in \eqref{intr} have been discussed in \cite{bfpla} using critical point theorem based on a pseudo-index related to the cohomological index. The problems with critical exponential nonlinearity has been discussed in \cite{dpj}, where authors have obtained the local multiplicity results for the problem having singular exponential nonlinearity. Also the asymptotic behavior of the connected unbounded branch of solutions is shown in the radial case. In the recent past the existence and multiplicity results have been obtained for the problems involving nonlocal fractional operators, see \cite{fbm, Gm1,Gm2,Gm3,Gm4, bffpl} and references therein. In \cite{fbm}, the authors considered the critical exponent problem \begin{equation*} (-\Delta )^su=\lambda u+u^{2^s_*-2}u, \text{ in }\Omega,\quad u=0 \text{ on }\partial \Omega, \end{equation*} where $\Omega \subset \mathbb{R}^n$, $s\in (0, 1)$ and $2^s_*=\frac{2n}{n-2s}$ is the fractional critical Sobolev exponent. Using the abstract critical point theorem, authors have generalized the results of Cerami, Fortuno and Struwe \cite{sfs} for the nonlocal setting. For the case of $p$-fractional operator, Perera Squassina et al. showed the multiplicity results in \cite{bffpl} using the idea of \cite{bfpla}. Recently the fractional nonlocal operators with exponential nonlinearity has been studied in \cite{ jpsree, DEA,Iasq, nonA}. In \cite{Iasq}, authors showed the existence of at least two solutions in the bounded domain of $\mathbb{R}$ by showing the Palais-smale condition in the suitable range. These results are fractional generalizations of results obtained in \cite{ruf}. In \cite{nonA}, authors obtained the existence result in whole $\mathbb{R}$ by assuming a vanishing potential which tackles the compactness issue. Further in \cite{DEA}, existence and multiplicity results were obtained with superlinear and convex-concave type sign changing critical exponential nonlinearity using idea of Nehari manifold by converting the nonlocal problem into local setting by harmonic extensions introduced in \cite{CS}. In \cite{DEA}, authors obtained the existence result for whole $\mathbb{R}$ using the suitable fractional version of Moser-Trudinger inequality. In this work we obtain the similar results as in \cite{fbm} for the nonlinearity having critical exponential growth in $\mathbb{R}$. The main difficulty in our problem is the lack of compactness. We use the Moser-Trundinger inequality, see \eqref{Pc} below, adapted from \cite{jpsree, Iasq} to get pass the limit. Now we define the function space where we look for the solutions of problem \eqref{ePl}. Also, we discuss the spectrum for the fractional Laplacian and some spectral properties related to our nonlocal operator. In \cite{sevo}, Servedei and Valdinoci introduced such spaces for the problems involving nonlocal operators. We adopt the same idea to define the linear space as follows: \begin{align*} X= \Big\{&u:\mathbb{R} \to\mathbb{R} \text{ is measurable, $u|_{(-1, 1)} \in L^2((-1, 1))$ and}\\ & \int_Q\frac{|u(x)- u(y)|^2}{|x-y|^2}\,dx\,dy<\infty\Big\}, \end{align*} where $Q=\mathbb{R}^{2}\setminus(\mathcal{C}(-1,1)\times \mathcal{C}(-1, 1))$ and $\mathcal{C}(-1,1)=\mathbb{R}\setminus (-1,1)$. The space $X$ is a normed linear space endowed with the norm \[ \|u\|_X^2 = \|u\|_2^2 +\int_{Q}\frac{|u(x)- u(y)|^{2}}{|x-y|^2}\,dx\,dy, \] where $\|\cdot \|_2$ denotes the $L^2((-1, 1))$ norm of $u$. Then we define the space $X_0$ as \[ X_0 = \{u\in X : u = 0 \text{ a.e. in } \mathbb{R}\setminus (-1, 1)\}. \] The space $X_0$ is a Hilbert space, see \cite{sevo}, with the inner product \begin{equation} \label{sclr} \langle u,v\rangle=\int_{\mathbb{R}\times \mathbb{R}} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^2}\,dx\,dy. \end{equation} The norm on $X_0$, induced from the scalar product in \eqref{sclr} is defined as \[ \|u\|_{X_0}=\Big(\int_{\mathbb{R}\times \mathbb{R}} \frac{|u(x)-u(y)|^{2}}{|x-y|^2}dx dy\Big)^{1/2}. \] Now we define an equivalent norm on $X_0$ as follows \[ \|u\|=\Big(\frac{1}{2\pi}\int_{\mathbb{R}\times \mathbb{R}} \frac{|u(x)-u(y)|^{2}}{|x-y|^2}dx dy\Big)^{1/2}. \] Then from \cite[Proposition 3.6]{hitch}, we have \[ \|u\|=\|(-\Delta)^{1/4}u\|_2^2. \] Problems of the type \eqref{ePl} are motivated from the following Moser-Trudinger inequality, see \cite{matrinazziint}. \noindent\textbf{Moser-Trudinger inequality:} For any $u\in X_0$, $e^{\alpha u^2}\in L^1((-1, 1))$ for all $\alpha>0$. Moreover for $\alpha\leq \pi$ \begin{equation}\label{mtine} \sup_{u\in X_0,\; \|u\|\leq 1}\int_{-1}^1 e^{\alpha u^2}dx<\infty. \end{equation} The above inequality is sharp in the sense that if $\alpha>\pi$ the integral in \eqref{mtine} is not finite. Now we discuss the eigenvalue problem of the type \[ (-\Delta)^{1/2}u=\lambda u\text{ in }(-1, 1),\quad u=0 \text{ in }\mathbb{R}\setminus(-1, 1). \] It is known that there exists a infinite sequence of eigenvalues $\lambda_1<\lambda_2\leq \dots \leq\lambda_n\leq \dots $ with $\lambda_n\to+\infty$ as $n\to \infty$. The eigenfunctions $\{\varphi_n\}$ corresponding to each eigenvalue $\lambda_n$ form an orthonormal basis for $L^2((-1, 1))$ and an orthogonal basis for $X_0$. For more details see the work of Servedei and Valdinoci in \cite{cpaa, dcds}. Let $\lambda\in \mathbb{R}$ be given. Denote $\lambda_*=\min\{\lambda_n: \lambda<\lambda_n\}$ then $\lambda_*=\lambda_n$ for some $n\in \mathbb N$ and $\lambda_*\geq \lambda_1$. Let $\lambda_*$ have multiplicity $m$, then \begin{gather*} \lambda_*=\lambda_1<\lambda_2\quad \text{if } n=1,\\ \lambda_{n-1}<\lambda_*=\lambda_n=\lambda_{n+1}=\dots =\lambda_{n+m-1}<\lambda_{n+m} \quad \text{if } n\geq 2. \end{gather*} For more details related to eigenvalue for the square root of the Laplacian in the interval $(-1, 1)$ see \cite{spectral}. Now we state our main result. \begin{theorem}\label{theom} Problem \eqref{ePl} has $m$ pairs of nontrivial solutions $\{-u_j^\lambda, u_j^\lambda\}$ for $\lambda$ lying in the sufficiently small left neighborhood of $\lambda_*$. Moreover $\|u_j^\lambda\|\to 0$ as $\lambda\to \lambda_*$. \end{theorem} To prove the Theorem \ref{theom} we use the following result, see \cite[Theorem 2.4]{crlit}. \begin{theorem}\label{cpthm} Let $H$ be a Hilbert space with the norm $\|\cdot\|$ and $I:H\to \mathbb{R}$ be a $C^1(H, \mathbb{R})$ functional satisfying the following conditions: \begin{itemize} \item[(A1)] $I(0)=0$ and $I(-u)=I(u)$; \item[(A2)] $I$ satisfies the Palais-Smale condition, in short $(PS)_c$, for $c\in (0, \beta)$ for some $\beta>0$; \item[(A3)] There exists closed subspaces $V, W$ of $H$ and constants $\rho, \delta, \eta$ with $\delta<\eta<\beta$ such that \begin{itemize} \item[(i)] $ I(u)\leq \eta$ for all $u\in W$\smallskip \item[(ii)] $I(u)\geq \delta$ for any $u\in V$ with $\|u\|=\rho$\smallskip \item[(iii)] $codim (V)<\infty$ and $dim W\geq codim V$. \end{itemize} \end{itemize} Then there exists at least $\dim W-\operatorname{codim} V$ pairs of critical points of $I$ with critical values belonging to the interval $[\delta, \eta]$. \end{theorem} \section{Bifurcation and multiplicity result} In this section, we define the variational functional related to the problem and we show that the functional satisfies the requirement of critical point Theorem \ref{cpthm}. A function $u\in X_0$ is called a weak solution of \eqref{ePl} if \begin{align*} &{\frac{1}{2\pi}}\int_{\mathbb{R}\times \mathbb{R}} \frac{(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^2}\,dx\,dy\\ &-\lambda\int_{-1}^1u(x)\phi(x)dx-\int_{-1}^1g(u(x))\phi(x)dx=0 \end{align*} for all $\phi\in X_0$. The variational functional $I_\lambda: X_0\to \mathbb{R}$ associated with \eqref{ePl} is defined as \begin{equation*} I_\lambda(u)=\frac{1}{2}\|u\|^2-\frac{\lambda}{2}\int_{-1}^1 |u|^2dx -\int_{-1}^1 G(u)dx, \end{equation*} where $G(t)= \int_0^t g(s)ds$ is the primitive of $g$. It is clear that the functional $I_\lambda \in C^1(X_0, \mathbb{R})$ and critical points of the functional $I_\lambda$ are the solutions of \eqref{ePl}. From the definition, $I_\lambda(0)=0$ and from $g$ being an odd function, $I_\lambda(-u)=I_\lambda(u)$. Hence the assumption $( A1)$ is satisfied. Now we show the geometric requirements as in (A3). We assume that \[ W= \operatorname{span} \{\varphi_1, \varphi _2,\dots \varphi_{n+m-1}\} \] and $V= X_0$ if $n=1$, otherwise \[ V=\{u\in X_0: \langle u,\varphi_j\rangle=0\;\;\forall\; 1\leq j\leq n-1\}. \] Then both $W$ and $V$ are closed subspaces of $X_0$ with dim $W=n+m-1$ and $\operatorname{codim}V=n-1$. Now take $u\in W$ then $u(x)= \sum_{j=1}^{n+m-1}\alpha_j\varphi_j(x)$ and \begin{equation*} \|u\|^2= \sum_{j=1}^{n+m-1}\alpha_j^2\|\varphi_j\|^2 = \sum_{j=1}^{n+m-1}\lambda_j\alpha_j^2 \leq \lambda_n \sum_{j=1}^{n+m-1}\alpha_j^2 =\lambda_n\|u\|_2^2=\lambda_*\|u\|_2^2. \end{equation*} Now using that $G(t)\geq\frac{1}{p}|t|^p$, for $u\in W$, we obtain \begin{align*} I_\lambda(u) &=\frac{1}{2}\|u\|^2-\frac{\lambda}{2}\|u\|_2^2 -\int_{-1}^1 G(u(x))dx \leq \frac{1}{2}(\lambda_*-\lambda)\|u\|_2^2 -\int_{-1}^1 G(u(x))dx\\ &\leq \frac{1}{2}(\lambda_*-\lambda)2^{\frac{p}{p-2}}\|u\|_p^2 -\frac{1}{p}\|u\|_p^p. \end{align*} Define $f(t)=\frac{1}{2}(\lambda_*-\lambda)2^{\frac{p}{p-2}}t^2-\frac{1}{p}t^p$, then $f(t)$ has a maximum at \[ t_0=\Big((\lambda_*-\lambda)2^{\frac{p}{p-2}}\Big)^\frac{1}{p-2}. \] Hence \[ I_\lambda(u)\leq \eta=(\frac{1}{2}-\frac{1}{p}) \Big((\lambda_*-\lambda)2^{\frac{p}{p-2}}\Big)^\frac{1}{p-2}. \] Note that we can make $\eta$ to be arbitrary small positive number by choosing $\lambda$ suitably close to $\lambda_*$. For the second part, we make use of the fact that for $u\in V$, we have $\|u\|^2\geq \lambda_*\|u\|_2^2$ and $G(t)\leq \frac{1}{p}e^{t^2}|t|^p$. Therefore \[ I_\lambda(u)\geq \big(\frac{\lambda_*-\lambda}{2\lambda_*}\big) \|u\|^2-\frac{1}{p}\int_{-1}^1|u|^pe^{u^2}dx \geq \big(\frac{\lambda_*-\lambda}{2\lambda_*}\big)\|u\|^2 -\frac{C}{p}\|u\|^p=\delta>0 \] for $\|u\|=\rho$ with sufficiently small value of $\rho>0$. Note that the second integral in the above expression is well defined by Moser-Trudinger inequality \eqref{mtine} for the choice of $\rho>0$ taken. Now we prove few Lemmas which are useful for showing compactness of Palais-Smale sequence. \begin{lemma}\label{bb} Every Palais-Smale sequence of $I_\lambda$ is bounded in $X_0$. \end{lemma} \begin{proof} Let $\{u_k\}$ be a Palais-Smale sequence that is \begin{equation}\label{Pc} | I_\lambda(u_k)|\leq C\quad\text{and}\quad \langle I'_{\lambda}(u_k),u_k\rangle \leq C\|u_k\|, \end{equation} for any $k\in\mathbb{N}$. Also using the fact that $G(t)\leq \frac{1}{p}g(t)t$, we obtain \begin{equation} \label{E1} \begin{aligned} I_{\lambda}(u_k)-\frac{1}{2}I'_{\lambda}(u_k)(u_k) &\geq\big(\frac{1}{2}-\frac{1}{p}\big)\int_{-1}^1g(u_k)u_kdx\\ &\geq\big(\frac{1}{2}-\frac{1}{p}\big)\int_{-1}^1|u_k|^pdx \geq C_0\|u_k\|^{p}. \end{aligned} \end{equation} On the other hand, from \eqref{Pc}, we obtain \begin{equation}\label{E2} I_{\lambda}(u_k)-\frac{1}{2}\langle I'_{\lambda}(u_k)(u_k)\rangle \leq C(1+\|u_k\|). \end{equation} Now, from \eqref{E1} and \eqref{E2}, we obtain $ \|u_k\|^p\leq C(1+\|u_k\|)$, which implies that sequence $\{u_k\}$ is bounded in $X_0$. \end{proof} \begin{lemma}\label{gcc} Let $\{u_k\}$ be a Palais-Smale sequence for $I_\lambda$ then $g(u_k)\to g(u)$ in $L^1((-1, 1))$ for some $u\in X_0$. \end{lemma} \begin{proof} From Lemma \ref{bb}, we obtain that $\{u_k\}$ is bounded, therefore \begin{equation}\label{relat} \|u\|\leq C,\quad \int_{-1}^1 g(u_k)u_kdx \leq C,\quad \int_{-1}^1 G(u_k)dx \leq C. \end{equation} So there exists $u\in X_0$ such that $u_k\rightharpoonup u $ in $X_0$, $u_k\to u$ in $L^\gamma ((-1, 1))$ for all $\gamma>1$ and $u_k(x)\to u(x)$ a.e. in $(-1, 1)$. Now the proof follows from Lebesgue dominated convergence theorem, see \cite[Lemma 2.1]{ruf}. \end{proof} Now we state a version of higher integrability Lemma. Proof is adapted from \cite[Lemma 4.1]{DEA}. \begin{lemma}\label{lio} Let $\{u_k\}$ be a sequence in $X_0$ with $\|u_k\|=1$ and $u_k \rightharpoonup u$ weakly in $X_0$. Then for any $r$ such that $10$ small enough and $\mu>1$ close to $1$, we have \[ \alpha r \mu (1+\epsilon) (1-\|u\|^2) <\pi \] and the proof follows from \eqref{mtine}. \end{proof} Now we show a compactness result. The proof follows closely \cite[Proposition 2.1]{ruf}. \begin{proposition} \label{prop1} Let $\{u_k\}$ be a Palais-Smale sequence, that is \begin{equation}\label{psdef} I_\lambda(u_k)=c+o_k(1) \quad \text{and}\quad I'_\lambda(u_k)=o_k(1) \end{equation} then it has a strongly convergent subsequence for $c\in (0, \frac{\pi}{2})$. \end{proposition} \begin{proof} From Lemma \ref{bb}, there exists $u\in X_0$ such that $u_k\rightharpoonup u $ in $X_0$, $u_k\to u$ in $L^\gamma ((-1, 1))$ for all $\gamma>1$ and $u_k(x)\to u(x)$ a.e. in $(-1, 1)$. Following the Lebesgue dominated convergence theorem and the relation \eqref{relat}, we have $G(u_k)\to G(u)$ in $L^1((-1, 1))$ and \begin{equation}\label{combine} \begin{gathered} \lim_{k\to\infty}\|u_k\|^2=2c+\lambda\|u\|_2^2+2\int_{-1}^1 G(u)dx,\\ \lim_{k\to \infty}\int_{-1}^1 g(u_k)u_kdx=2c+2\int_{-1}^1 G(u)dx. \end{gathered} \end{equation} Now from the weak convergence of $u_k$ in $X_0$, strong convergence in $L^2((-1, 1))$ and the Lemma \ref{gcc}, $u$ solves the problem \eqref{ePl}. Moreover, \begin{equation*} \|u\|^2-\lambda\|u\|_2^2=\int_{-1}^1g(u)u\,dx\geq2\int_{-1}^1 G(u)\,dx. \end{equation*} Therefore $I_\lambda(u)\geq 0$. Now we divide the proof into three cases. \smallskip \noindent\textbf{Case 1: $c=0$.} In this case $\lim\inf I_\lambda(u_k)=0$. Therefore $\|u_k\|\to \|u\|$, and the proof is complete. \smallskip \noindent\textbf{Case 2: $c\neq0, u=0$.} As $u=0$, so from \eqref{combine} we obtain $ \lim_{k\to\infty} \|u_k\|=2c$. Hence for every $\epsilon>0$ and for large $k$, $\|u_k\|\leq 2c+\epsilon$. Now for some $q>1$ and close to $1$, we have the estimate \begin{equation} \label{mtd} \begin{aligned} \int_{-1}^1|g(u_k)|^qdx &=\int_{-1}^1|u_k|^{qp}e^{qu_k^2}dx\\ &\leq\|u_k\|^{qp}\Big(\int_{-1}^1e^{qlu_k^2}dx\Big)^{1/l}\\ &\leq C\Big(\sup_{\|v\|\leq 1}\int_{-1}^1e^{lq\|u_k\|^2v^2}dx\Big)^{1/l}<\infty \end{aligned} \end{equation} for $c<\pi/2$, $l>1$ close to 1 and $\epsilon$ sufficiently small. Now from \eqref{psdef}, we obtain \begin{equation} \label{ps22} \big| \|u_k\|^2-\lambda\|u_k\|_2^2-\int_{-1}^1g(u_k)u_k dx\big| \leq \epsilon _k\|u_k\|. \end{equation} Now from the estimate in \eqref{mtd} inequality \eqref{ps22} and $u_k\to 0$ in $L^\gamma((-1,1))$ we obtain $\|u_k\|\to 0$, which is the contradiction from \eqref{combine} as $c\neq 0$. \smallskip \noindent\textbf{Case 3: $c\neq 0$, $u\neq 0$.} In this case we claim that $I_\lambda(u)=c$. If the claim is proved then from \eqref{combine}, $ \lim_{k\to \infty}\|u_k\|=\|u\|$ and hence the proof of the Lemma. Suppose not, then $I_\lambda(u)