On a class of semipositone problems with singular Trudinger-Moser nonlinearities

Shiqiu Fu,Kanishka Perera

doi:10.3934/dcdss.2020452

Abstract

<p style='text-indent:20px;'>We prove the existence of positive solutions for a class of semipositone problems with singular Trudinger-Moser nonlinearities. The proof is based on compactness and regularity arguments.

Highlights

Let Ω be a bounded domain in RN, N ≥ 2 and let f be a Caratheodory function on Ω × [0, ∞)
We prove the existence of positive solutions for a class of semipositone problem with singular Trudinger-Moser nonlinearities
The purpose of the present paper is to study a class of semipositone problems with singular exponential nonlinearities in dimension N = 2

Summary

Introduction

Let Ω be a bounded domain in RN , N ≥ 2 and let f be a Caratheodory function on Ω × [0, ∞). It is notoriously difficult to find positive solutions of this class of problems due to the fact that u = 0 is not a subsolution Our problem is critical with respect to this embedding and the variational functional associated with this problem lacks compactness, which is an additional difficulty in finding solutions. There exists a μ∗ > 0 such that for all 0 < μ < μ∗, problem (1.1) has a solution uμ We note that this result does not follow from standard arguments based on the maximum principle since g(0) is not assumed to be nonnegative. Our proof is based on regularity arguments and will be given, after establishing a suitable compactness property of an associated variational functional Our proof is based on regularity arguments and will be given in Section 3, after establishing a suitable compactness property of an associated variational functional

In this section we consider the modified problem

Next we claim that

Since u is a nontrivial solution of the problem