A resonance problem for non-local elliptic operators

Alessio Fiscella, Raffaella Servadei, Enrico Valdinoci

Research output: Contribution to journalArticlepeer-review

37 Citations (Scopus)

Abstract

In this paper we consider a resonance problem driven by a non-local integrodifferential operator LK 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) in u = 0 in Rn n ; when λ is an eigenvalue of the related non-homogenous linear problem with homogeneous Dirichlet boundary data. Here the parameter s 2 (0; 1) is fixed, is an open bounded set of Rn, n > 2s, with Lipschitz boundary, a is a Lipschitz continuous function, while f is a suffciently 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;.

Original languageEnglish
Pages (from-to)411-431
Number of pages21
JournalZeitschrift fur Analysis und ihre Anwendung
Volume32
Issue number4
DOIs
Publication statusPublished - 22 Oct 2013
Externally publishedYes

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