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 (-∆)su = λƒ(x, u) in Ω, u = 0 on ℝN \ Ω, where Ω is an open bounded subset in ℝN with Lipschitz boundary, λ is a real parameter, 0 < s < 1, N > 2s, and ƒ : Ω x ℝ → 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 λ for which the problem above with discontinuous nonlinearities admits at least three nontrivial weak solutions by applying two recent three-critical-points theorems.