Multiplicity results for fractional Laplace problems with critical growth

Abstract

This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator $(-\Delta )^s$ and involving a critical Sobolev term. In particular, we consider

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^su=\gamma \left| u\right| ^{2^*-2}u+f(x,u) &{} \text{ in } \Omega \\ u=0 &{} \text{ in } \mathbb {R}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$

where $\Omega \subset \mathbb {R}^n$ is an open bounded set with continuous boundary, $n>2s$ with $s\in (0,1)$, $\gamma$ is a positive real parameter, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and f is a Carathéodory function satisfying different subcritical conditions. For this problem we prove two different results of multiple solutions in the case when f is an odd function. When f has not any symmetry it is still possible to get a multiplicity result: we show that the problem under consideration admits at least two solutions of different sign.