Positive solutions for a class of singular Dirichlet problems

TL;DR

This paper investigates positive solutions for a singular elliptic Dirichlet problem with superlinear nonlinearities, identifying a critical parameter value that determines the existence of solutions.

Contribution

It establishes the existence of a critical parameter for positive solutions in a singular Dirichlet problem with superlinear terms, extending previous work without AR-condition assumptions.

Findings

For λ > λ*, there are two positive solutions.

For λ < λ*, no positive solutions exist.

The critical case λ = λ* remains an open problem.

Abstract

We consider a Dirichlet elliptic problem driven by the Laplacian with singular and superlinear nonlinearities. The singular term appears on the left-hand side while the superlinear perturbation is parametric with parameter $\lambda>0$ and it need not satisfy the AR-condition. Having as our starting point the work of Diaz-Morel-Oswald (1987), we show that there is a critical parameter value $\lambda_*$ such that for all $\lambda>\lambda_*$ the problem has two positive solutions, while for $\lambda<\lambda_*$ there are no positive solutions. What happens in the critical case $\lambda = \lambda_*$ is an interesting open problem.