A critical fractional Laplace equation in the resonant case

Abstract

In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities $$ \cases (-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\text{in }} \Omega,\\ u=0 & {\text{in }} \mathbb{R}^n\setminus \Omega, \endcases $$ started in the recent papers \cite{13}, \cite{17}-\cite{19}. Here $s\in (0,1)$ is a fixed parameter, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive constant, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $\RR^n$, $n> 2s$, with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when $n\not=4s$ and $\lambda$ is an eigenvalue of the operator $(-\Delta)^s$ with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a non-trivial solution, so that with the results obtained in \cite{13}, \cite{17}-\cite{19}, we are able to show that this critical problem admits a nontrivial solution provided \roster \item"$\bullet$" $n> 4s$ and $\lambda> 0$, \item"$\bullet$" $n=4s$ and $\lambda> 0$ is different from the eigenvalues of $(-\Delta)^s$, \item"$\bullet$" $2s< n< 4s$ and $\lambda> 0$ is sufficiently large. \endroster In this way we extend completely the famous result of Brezis and Nirenberg (see \cite{4}, \cite{5}, \cite{9}, \cite{23}) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.