On the solvability of resonance problems for nonlocal elliptic equations

In this article, we consider the following problem: $$\quad \left\{ \begin{array}{lr} \quad (-\Delta)^s u = \alpha u^+ -\beta u^{-} + f(u) + h \; \text{in}\;\Omega \quad \quad \quad \quad u =0 \; \text{on}\; \mathbb{R}^n\setminus \Omega, \end{array} \right.$$ 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 Fu\v{c}ik spectrum. Second, the resonance case, where $(\alpha,\beta)$ is an element of the Fu\v{c}ik 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.