A Resonance Problem for Non-Local Elliptic Operators

In this paper we consider a resonance problem driven by a non-local integrodifferential operator 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 \{ {alignedat}{2} (-Δ)^s u&=λ a(x)u+f(x,u)& &{in } Ω\\ u&=0& &{in } R^n Ω, {alignedat} . when λ is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data. Here the parameter s∈ (0,1) is fixed, Ω is an open bounded set of 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 non-local setting some results, already known in the literature in the case of the Laplace operator -Δ .