Positive solutions of semipositone elliptic problems with critical Trudinger-Moser nonlinearities

TL;DR

This paper establishes the existence of positive solutions for a class of semipositone elliptic problems involving the critical Trudinger-Moser nonlinearity, using uniform a priori estimates and compactness arguments.

Contribution

It provides the first proof of positive solutions for semipositone problems with critical Trudinger-Moser nonlinearities, including the case N=2.

Findings

Existence of positive solutions proven for the problem.

Uniform $C^{1,eta}$ a priori estimates established.

Method adaptable to more general semipositone problems.

Abstract

We prove the existence of a positive solution to a semipositone $N$-Laplacian problem with a critical Trudinger-Moser nonlinearity. The proof is based on obtaining uniform $C^{1,\alpha}$ a priori estimates via a compactness argument. Our result is new even in the semilinear case $N = 2$, and our arguments can easily be adapted to obtain positive solutions of more general semipositone problems with critical Trudinger-Moser nonlinearities.