Nonresonance and Resonance Problems for Nonlocal Elliptic Equations with Respect to the Fučik Spectrum

Abstract

In this article, we consider the following problem \[ \begin{cases} (-\Delta)^s u = \alpha u^+ - \beta u^{-} + f(u) + h &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\mathbb{R}^n \setminus \Omega$}, \end{cases} \] where $\Omega \subset \mathbb{R}^n$ is a bounded domain with Lipschitz boundary, $n \gt 2s$, $0 \lt s \lt 1$, $(\alpha,\beta) \in \mathbb{R}^2$, $f \colon \mathbb{R} \to \mathbb{R}$ 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.