The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (327 KB) | Metadata | Table of Contents | ZAA summary
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