Existence, nonexistence and multiplicity of bounded solutions to a nonlinear BVP associated to the fractional Laplacian

Abstract

We deal with the boundary value problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u(x)= \lambda f(u(x)), & x\in \Omega ,\\ u(x)=0, & x\in \mathbb {R}^N \setminus \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \) is an open and bounded subset of \(\mathbb {R}^N\) with smooth boundary, \((-\Delta )^s\), \(s\in (0,1)\) denotes the fractional Laplacian, \(\lambda \ge 0\) and f is locally Lipschitz and continuous. We provide necessary and sufficient conditions on f to ensure existence and multiplicity of bounded solutions between two zeros of f.