The effect of the domain topology on the number of solutions of fractional Laplace problems

Abstract

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^n$$ is an open bounded set with continuous boundary, $$n>2s$$ with $$s\in (0,1),(-\Delta )^{s}$$ is the fractional Laplacian operator, $$\mu $$ is a positive real parameter, $$q\in [2, 2^*_s)$$ and $$2^*_s=2n/(n-2s)$$ is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of $$\Omega $$ . Precisely, we show that the problem has at least $$cat_{\Omega }(\Omega )$$ nontrivial solutions, provided that $$q=2$$ and $$n\geqslant 4s$$ or $$q\in (2, 2^*_s)$$ and $$n>2s(q+2)/q$$ , extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.