Existence and nonexistence of positive solutions for singular (p,q)-equations with superdiffusive perturbation

TL;DR

This paper investigates the conditions under which positive solutions exist or do not exist for a nonlinear Dirichlet problem involving the $(p,q)$-Laplacian with singular and superdiffusive reaction terms, depending on a parameter.

Contribution

It provides new existence and nonexistence results for positive solutions of $(p,q)$-Laplacian equations with singular and superdiffusive effects based on parameter values.

Findings

Positive solutions exist for certain parameter ranges.

No positive solutions for other parameter ranges.

The results depend on the interplay between singularity and superdiffusive terms.

Abstract

We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian and with a reaction which is parametric and exhibits the combined effects of a singular term and of a superdiffusive one. We prove an existence and nonexistence result for positive solutions depending on the value of the parameter $\lambda \in \overset{\circ}{\mathbb{R}}_+=(0,+\infty)$.