Positive solutions for a class of singular Dirichlet problems

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 ${\lambda }_{\ast }$ such that for all $\lambda >{\lambda }_{\ast }$ the problem has two positive solutions, while for $\lambda <{\lambda }_{\ast }$ there are no positive solutions. What happens in the critical case $\lambda ={\lambda }_{\ast }$ is an interesting open problem.

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