Multiple solutions for discontinuous elliptic problems involving the fractional Laplacian

Abstract

In this article, we establish the existence of three weak solutions for elliptic equations associated to the fractional Laplacian $$\displaylines{ (-\Delta)^s u = \lambda f(x,u) \quad \text{in } \Omega,\cr u= 0\quad \text{on } \mathbb{R}^N\setminus\Omega, }$$ where $\Omega$ is an open bounded subset in $\mathbb{R}^{N}$ with Lipschitz boundary, $\lambda$ is a real parameter, 02s, and $f:\Omega\times\mathbb{R} \to \mathbb{R}$ is measurable with respect to each variable separately. The main purpose of this paper is concretely to provide an estimate of the positive interval of the parameters $\lambda$ for which the problem above with discontinuous nonlinearities admits at least three nontrivial weak solutions by applying two recent three-critical-points theorems.

Keywords

• Fractional Laplacian
• three-critical-points theorem
• multiple solutions