Positive solutions for singular anisotropic $(p,q)$-equations

TL;DR

This paper investigates positive solutions for a class of anisotropic $(p,q)$-equations with singular and superlinear terms, establishing bifurcation results, existence of minimal solutions, and their properties.

Contribution

It introduces a bifurcation theorem for anisotropic $(p,q)$-equations with singular and superlinear reactions, and analyzes the minimal positive solution's behavior.

Findings

Bifurcation diagram for positive solutions as parameter varies

Existence of a minimal positive solution

Monotonicity and continuity of the minimal solution map

Abstract

In this paper we consider a Dirichlet problem driven by an anisotropic $(p,q)$-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.