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On the solvability of resonance problems for nonlocal elliptic equations

In this article, we consider the following problem: $$ \quad \left\{ \begin{array}{lr} \quad (-Δ)^s u = αu^+ -βu^{-} + f(u) + h \; \text{in}\;Ω\quad \quad \quad \quad u =0 \; \text{on}\; \mathbb{R}^n\setminus Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^n$ is a bounded domain with Lipschitz boundary, $n> 2s$, $0<s<1$, $(α, β) \in \mathbb{R}^2$, $f: \mathbb{R}\to \mathbb{R}$ is a bounded and continuous function and $h\in L^2(Ω)$. We prove the existence results in two cases: First, the nonresonance case, where $(α,β)$ is not an element of the Fučik spectrum. Second, the resonance case, where $(α,β)$ 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.