A resonance problem for non-local elliptic operators (Q380259)

Summary: In this paper we consider a resonance problem driven by a nonlocal integrodifferential operator \(\mathcal L_K\) with homogeneous Dirichlet boundary conditions. This problem has a variational structure and we find a solution for it using a 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 \[ (-\Delta)^s u=\lambda a(x)u+f(x,u) \;\;\text{in} \;\Omega \] \[ u=0 \;\;\text{in} \;\mathbb{R}^n\setminus \Omega, \] 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 subset of \(\mathbb{R}^n, n>2s\), with Lipschitz boundary, \(a\) is a Lipschitz continuous function, while \(f\) is a sufficiently smooth function. This existence theorem extends to the nonlocal setting some results already known in the literature in the case of the Laplace operator \(-\Delta\).