An existence result for anisotropic quasilinear problems

TL;DR

This paper proves the existence of solutions for a class of boundary degenerate quasilinear equations involving weighted p-Laplacian operators, under minimal growth conditions and the existence of sub- and supersolutions.

Contribution

It establishes an existence theorem for anisotropic quasilinear problems with degenerate or singular boundary conditions, extending previous results to more general nonlinearities.

Findings

Existence of solutions under minimal growth restrictions.

Applicability to problems with non-linearities satisfying certain boundary conditions.

Extension to weighted p-Laplacian operators with integrability assumptions.

Abstract

We study existence of solutions for a boundary degenerate (or singular) quasilinear equation in a smooth bounded domain under Dirichlet boundary conditions. We consider a weighted $p-${L}aplacian operator with a coefficient that is {locally bounded inside the domain and satisfying certain additional integrability assumptions}. Our main result applies for boundary value problems involving continuous non-linearities having no growth restriction, but provided the existence of a sub and a supersolution is guaranteed. As an application, we present an existence result for a boundary value pro\-blem with a non-linearity $f(u)$ satisfying $f(0) \leq 0$ and having $(p-1)-$sublinear growth at infinity.