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 $$ \begin{cases} (-\Delta)^s u - {\lambda}u = |u|^{2^* -2}u \; \mathrm{in} \; \Omega, \\ u=0 \; \mathrm{in} \; \mathbb{R}^n \setminus \Omega, \end{cases} $$ started in the recent papers [13], [17]-[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 $\mathbb R^n$, $n> 2s$, 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 [13], [17]-[19], we are able to show that this critical problem admits a nontrivial solution provided

$\bullet$ $n> 4s$ and $\lambda> 0$,

$\bullet$ $n=4s$ and $\lambda> 0$ is different from the eigenvalues of $(-\Delta)^s$,

$\bullet$ $2s< n< 4s$ and $\lambda> 0$ is sufficiently large.

In this way we extend completely the famous result of Brezis and Nirenberg (see [4], [5], [9], [23]) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.