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# Zeitschrift für Analysis und ihre Anwendungen

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Volume 32, Issue 4, 2013, pp. 411–431
DOI: 10.4171/ZAA/1492

Published online: 2013-09-19

A Resonance Problem for Non-Local Elliptic Operators

Alessio Fiscella[1], Raffaella Servadei[2] and Enrico Valdinoci[3]

(1) Università degli Studi di Milano, Italy
(2) Università della Calabria, Cosenza, Italy
(3) Università degli Studi di Milano, Italy

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 $-\Delta$.}

Keywords: Integrodifferential operators, fractional Laplacian, variational techniques, Saddle Point Theorem, Palais-Smale condition

Fiscella Alessio, Servadei Raffaella, Valdinoci Enrico: A Resonance Problem for Non-Local Elliptic Operators. Z. Anal. Anwend. 32 (2013), 411-431. doi: 10.4171/ZAA/1492