Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $

Silvia Frassu

doi:10.3934/cpaa.2019086

Article Contents

2019, Volume 18, Issue 4: 1847-1867. Doi: 10.3934/cpaa.2019086

This issue Previous Article Second order non-autonomous lattice systems and their uniform attractors Next Article Global attractors for a mixture problem in one dimensional solids with nonlinear damping and sources terms

Nonlinear Dirichlet problem for the nonlocal anisotropic operator $ L_K $

Silvia Frassu^,

Department of Mathematics and Computer Science, University of Cagliari, Viale L. Merello 92, 09123 Cagliari, Italy

* Corresponding author
* Corresponding author

Received: July 2018

Revised: September 2018

Published: January 2019

The author is member of GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica 'Francesco Severi').

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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 such as $ L^{\infty} $ 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 $ C^0 $-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the $ X(\Omega) $-topology.

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Mathematics Subject Classification: 35R09, 35R11, 47G20.

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