Positive solutions of singular elliptic problems with unbounded exponents and unbounded convection term

TL;DR

This paper proves the existence of positive solutions for a broad class of singular elliptic problems with unbounded exponents and convection terms, using sub- and super-solution methods and fixed point theorems.

Contribution

It introduces a novel approach to handle elliptic problems with unbounded exponents and convection terms, expanding the scope of solvable singular PDEs.

Findings

Existence of positive solutions under broad conditions

Application of sub- and super-solution method with Schauder's fixed point

Addresses previously unexplored unbounded convection terms

Abstract

In this paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-\Delta_p u = \frac{\lambda}{u^{\alpha}} + f(x,u,\nabla u)$$ in a bounded, smooth domain $\Omega$. The convection term has exponents with no upper limitations neither in $u$ nor in $\nabla u$. This is somewhat unexpected and rare. So, we address a wide range of problems not yet contained in the literature. The solution of the problem combines the definition of an auxiliary problem, the method of sub- and super-solution and Schauder's fixed point theorem.