A Resonance Problem for Non-Local Elliptic Operators

Abstract

In this paper we consider a resonance problem driven by a non-local integrodifferential operator ${\mathcal{L}}_K$ with homogeneous Dirichlet boundary conditions. This problem has a variational structure and we find a solution for it using the Saddle Point Theorem. We prove this result for a general integrodifferential operator of fractional type and from this, as a particular case, we derive an existence theorem for the following fractional Laplacian equation

\[ \left\{ \begin{alignedat}{2} (-\Delta)^s u&=\lambda a(x)u+f(x,u)& \quad&{\mbox{in }} \Omega\\ u&=0& &{\mbox{ in }} \mathbb{R}^n\setminus \Omega, \end{alignedat} \right. \]

when $\lambda$ is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data. Here the parameter $s\in (0,1)$ is fixed, $\Omega$ is an open bounded set of \( \RR^n \), $n>2s$, with Lipschitz boundary, $a$ is a Lipschitz continuous function, while $f$ is a sufficiently smooth function. This existence theorem extends to the non-local setting some results, already known in the literature in the case of the Laplace operator $-\Delta$.}