TY - JOUR
T1 - A resonance problem for non-local elliptic operators
AU - Fiscella, Alessio
AU - Servadei, Raffaella
AU - Valdinoci, Enrico
PY - 2013/10/22
Y1 - 2013/10/22
N2 - 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;.
AB - 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;.
KW - Fractional laplacian
KW - Integrodifferential operators
KW - Palais-Smale condition
KW - Saddle point theorem
KW - Variational techniques
UR - http://www.scopus.com/inward/record.url?scp=84885786892&partnerID=8YFLogxK
U2 - 10.4171/ZAA/1492
DO - 10.4171/ZAA/1492
M3 - Article
AN - SCOPUS:84885786892
SN - 0232-2064
VL - 32
SP - 411
EP - 431
JO - Zeitschrift fur Analysis und ihre Anwendung
JF - Zeitschrift fur Analysis und ihre Anwendung
IS - 4
ER -