A class of semipositone $p$-Laplacian problems with a critical growth reaction term

TL;DR

This paper establishes the existence of positive solutions for a class of semipositone p-Laplacian problems with critical growth, using a priori estimates and concentration compactness, extending results even to the semilinear case.

Contribution

It introduces new existence results for semipositone p-Laplacian problems with critical growth, employing novel a priori estimates and compactness techniques.

Findings

Existence of ground state positive solutions proven.

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

Results extend to the semilinear case $p=2$.

Abstract

We prove the existence of ground state positive solutions for a class of semipositone $p$-Laplacian problems with a critical growth reaction term. The proofs are established by obtaining crucial uniform $C^{1,\alpha}$ a priori estimates and by concentration compactness arguments. Our results are new even in the semilinear case $p = 2$.