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;.

Fiscella, A., Servadei, R., Valdinoci, E. (2013). A resonance problem for non-local elliptic operators. ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN, 32(4), 411-431 [10.4171/ZAA/1492].

A resonance problem for non-local elliptic operators

A. Fiscella;
2013

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;.
Articolo in rivista - Articolo scientifico
integrodifferential operators, fractional Laplacian, variational techniques, Saddle Point Theorem, Palais-Smale condition
English
411
431
21
Fiscella, A., Servadei, R., Valdinoci, E. (2013). A resonance problem for non-local elliptic operators. ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN, 32(4), 411-431 [10.4171/ZAA/1492].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/338152
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