Quasilinear Dirichlet systems with competing operators and convection

TL;DR

This paper addresses the existence of solutions for a quasi-linear Dirichlet system involving competing operators and convection, overcoming challenges posed by the lack of standard mathematical structures.

Contribution

It introduces an approximation method combined with Brouwer's fixed point theorem to establish solution existence without relying on ellipticity or variational frameworks.

Findings

Existence of solutions proven for complex quasi-linear systems

Method applicable despite absence of ellipticity and variational structure

Provides a new approach for systems with competing operators and convection

Abstract

In this paper, we consider a quasi-linear Dirichlet system with possible competing $(p,q)$-Laplacians and convections. Due to the lack of ellipticity, monotonicity, and variational structure, the standard approaches to the existence of weak solutions cannot be adopted. Nevertheless, through an approximation procedure and a corollary of Brouwer's fixed point theorem we show that the problem admits a solution in a suitable sense.