Existence, nonexistence and multiplicity of positive solutions for singular quasilinear problems

TL;DR

This paper investigates the existence, nonexistence, and multiplicity of positive solutions for a singular quasilinear problem with a parametric superlinear perturbation, using variational and sub-supersolution methods.

Contribution

It provides new results on positive solutions for singular quasilinear problems with arbitrary and subcritical superlinear growth, employing sub-supersolution and Mountain Pass techniques.

Findings

Existence of at least one positive solution with arbitrary superlinear growth.

Existence of a second positive solution under subcritical growth conditions.

Identification of parameter ranges for solution existence and multiplicity.

Abstract

In the present paper we deal with a quasilinear problem involving a singular term and a parametric superlinear perturbation. We are interested in the existence, nonexistence and multiplicity of positive solutions as the parameter $\lambda>0$ varies. In our first result, the superlinear perturbation has an arbitrary growth and we obtain the existence of a solution for the problem by using the sub-supersolution method. For the second result, the superlinear perturbation has subcritical growth and we employ the Mountain Pass Theorem to show the existence of a second solution.