A Brezis-Nirenberg result for non-local critical equations in low dimension

Abstract

The present paper is devoted to the study of the following non-local fractional equation involving critical nonlinearities\begin{eqnarray}(-\Delta)^s u-\lambda u=|u|^{2^*-2}u, in \Omega \\u=0, in R^n\setminus \Omega,\end{eqnarray} where $s\in (0,1)$ is fixed, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive parameter, $2^*$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $R^n$, $n>2s$, with Lipschitz boundary. In the recent papers [14, 18, 19] we investigated the existence of non-trivial solutions for this problem when $\Omega$ is an open bounded subset of $R^n$ with $n\geq 4s$ and, in this framework, we prove some existence results.Aim of this paper is to complete the investigation carried on in [14, 18, 19], by considering the case when $2s < n < 4s$. In this context, we prove an existence theorem for our problem, which may be seen as a Brezis-Nirenberg type result in low dimension.In particular when $s=1$ (and consequently $n=3$) our result is the classical result obtained by Brezis and Nirenberg in the famous paper [4].In this sense the present work may be considered as the extensionof some classical results for the Laplacian to the case of non-local fractional operators.