On the solvability of resonance problems for nonlocal elliptic equations

TL;DR

This paper investigates the existence of solutions for a nonlocal elliptic equation involving the fractional Laplacian, addressing both resonance and nonresonance cases, and extends classical results from the Laplace operator to nonlocal operators.

Contribution

It provides new existence results for resonance and nonresonance problems involving the fractional Laplacian, extending classical Laplace operator results to nonlocal operators.

Findings

Existence of solutions in nonresonance case proven.

Existence of solutions in resonance case proven.

Application of Saddle point Theorem to nonlocal equations.

Abstract

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.