Nonlinear Dirichlet problem for the nonlocal anisotropic operator L_K

2018/05/29 by Frassu, Silvia
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.11549

Abstract

In this paper we study an equation driven by a nonlocal anisotropic operator with homogeneous Dirichlet boundary conditions. We find at least three non trivial solutions: one positive, one negative and one of unknown sign, using variational methods and Morse theory. We present some results about regularity of solutions as L^∞-bound and Hopf's lemma, for the latter we first consider a non negative nonlinearity and then a strictly negative one. Moreover, we prove that, for the corresponding functional, local minimizers with respect to a C⁰-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the X(Ω)-topology.

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• The Dirichlet problem for nonlocal operators
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• Nonlocal operators in divergence form and existence theory for integrable data
• Multiplicity results for non-local operators of elliptic type
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