Journal
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Volume 389, Issue 2, Pages 887-898Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jmaa.2011.12.032
Keywords
Mountain Pass Theorem; Variational techniques; Integrodifferential operators; Fractional Laplacian
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Funding
- MIUR
- ERC
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The purpose of this paper is to study the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a non-trivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. We prove this result for a general integrodifferential operator of fractional type and, as a particular case, we derive an existence theorem for the fractional Laplacian, finding non-trivial solutions of the equation {(-Delta)(s)u = f (x, u) in Omega, u = 0 in R-n \Omega. As far as we know, all these results are new. (C) 2011 Elsevier Inc. All rights reserved.
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