The Brezis-Nirenberg result for the fractional Laplacian

Abstract:

The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation \[ \left \{ \begin {array}{ll} (-\Delta )^s u-\lambda u=|u|^{2^*-2}u & {\mbox { in }} \Omega ,\\ u=0 & {\mbox { in }} \mathbb {R}^n\setminus \Omega , \end {array} \right . \] where $(-\Delta )^s$ is the fractional Laplace operator, $s\in (0,1)$, $\Omega$ is an open bounded set of $\mathbb {R}^n$, $n>2s$, with Lipschitz boundary, $\lambda >0$ is a real parameter and $2^*=2n/(n-2s)$ is a fractional critical Sobolev exponent.

In this paper we first study the problem in a general framework; indeed we consider the equation \[ \left \{ \begin {array}{ll} \mathcal L_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 & \mbox {in } \Omega ,\\ u=0 & \mbox {in } \mathbb {R}^n\setminus \Omega , \end {array}\right .\] where $\mathcal L_K$ is a general non-local integrodifferential operator of order $s$ and $f$ is a lower order perturbation of the critical power $|u|^{2^*-2}u$. In this setting we prove an existence result through variational techniques.

Then, as a concrete example, we derive a Brezis-Nirenberg type result for our model equation; that is, we show that if $\lambda _{1,s}$ is the first eigenvalue of the non-local operator $(-\Delta )^s$ with homogeneous Dirichlet boundary datum, then for any $\lambda \in (0, \lambda _{1,s})$ there exists a non-trivial solution of the above model equation, provided $n\geqslant 4s$. In this sense the present work may be seen as the extension of the classical Brezis-Nirenberg result to the case of non-local fractional operators.