Anisotropic $(p,q)$-equations with gradient dependent reaction

TL;DR

This paper studies a complex anisotropic differential equation involving gradient-dependent reactions, employing advanced mathematical techniques to prove the existence of positive smooth solutions.

Contribution

It introduces a novel approach using the frozen variable method and fixed point theorem for anisotropic $(p,q)$-Laplacian problems with gradient reactions.

Findings

Existence of positive smooth solutions established.

Method applicable to anisotropic $(p,q)$-Laplacian with gradient-dependent reactions.

Overcomes challenges posed by non-variational structure.

Abstract

We consider a Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and a reaction with gradient dependence (convection). The presence of the gradient in the source term excludes from consideration a variational approach in dealing with the qualitative analysis of this problem with unbalanced growth. Using the frozen variable method and eventually a fixed point theorem, the main result of this paper establishes that the problem has a positive smooth solution.