On the solvability of resonance problems for nonlocal elliptic equations

Abstract

In this article, we consider the following problem: $$\{\begin{array}{l}(-\Delta {)}^{s}u=\alpha u^{+}-\beta u^{-}+f(u)+h\text{in}\Omega u=0\text{on}{\mathbb{R}}^n\setminus \Omega ,\end{array}$$ where $\Omega \subset {\mathbb{R}}^n$ is a bounded domain with Lipschitz boundary, $n>2s$, $0<s<1$, $(\alpha ,\beta )\in {\mathbb{R}}^2$, $f:\mathbb{R}\to \mathbb{R}$ is a bounded and continuous function and $h\in L^2(\Omega )$. We prove the existence results in two cases: First, the nonresonance case, where $(\alpha ,\beta )$ is not an element of the Fučik spectrum. Second, the resonance case, where $(\alpha ,\beta )$ is an element of the Fučik spectrum. Our existence results follows as an application of the Saddle point Theorem. It extends some results, well known for Laplace operator, to the nonlocal operator.