A Resonance Problem for Non-Local Elliptic Operators

  • Alessio Fiscella

    Università degli Studi di Milano, Italy
  • Raffaella Servadei

    Università della Calabria, Cosenza, Italy
  • Enrico Valdinoci

    Università degli Studi di Milano, Italy


In this paper we consider a resonance problem driven by a non-local integrodifferential operator LK\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

{(Δ)su=λa(x)u+f(x,u)\mboxinΩu=0\mboxinRnΩ,\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 λ\lambda is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data. Here the parameter s(0,1)s\in (0,1) is fixed, Ω\Omega is an open bounded set of \RRn\RR^n, n>2sn>2s, with Lipschitz boundary, aa is a Lipschitz continuous function, while ff 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-\Delta.}

Cite this article

Alessio Fiscella, Raffaella Servadei, Enrico Valdinoci, A Resonance Problem for Non-Local Elliptic Operators. Z. Anal. Anwend. 32 (2013), no. 4, pp. 411–431

DOI 10.4171/ZAA/1492