Nonlinear Dirichlet problem for the nonlocal anisotropic operator $L_K$

TL;DR

This paper investigates a nonlocal anisotropic Dirichlet problem, establishing the existence of multiple solutions, analyzing their regularity, and exploring the variational properties of the associated functional.

Contribution

It introduces new existence results for multiple solutions of a nonlocal anisotropic PDE using variational and Morse theory methods.

Findings

Existence of at least three nontrivial solutions: positive, negative, and indefinite.

Regularity results including $L^{ abla}-$bounds and Hopf's lemma for solutions.

Local minimizers in a weighted $C^0$-topology are also minimizers in the $X(rac{}{}$-topology.

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 $L^{\infty}$-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.