Anisotropic singular Neumann equations with unbalanced growth

TL;DR

This paper investigates positive solutions for a nonlinear anisotropic Neumann problem involving the $(p,q)$-Laplacian with singular and superlinear terms, establishing bifurcation results and properties of minimal solutions.

Contribution

It introduces a bifurcation analysis for anisotropic Neumann problems with singular and superlinear reactions, using topological and variational methods.

Findings

Existence of positive solutions depending on the parameter $mbda$

Identification of minimal positive solutions $u_mbda^*$

Monotonicity and continuity of the solution map $mbda rr u_mbda^*$

Abstract

We consider a nonlinear parametric Neumann problem driven by the anisotropic $(p,q)$-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and comparison techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter $\lambda$ varies. We also show the existence of minimal positive solutions $u_\lambda^*$ and determine the monotonicity and continuity properties of the map $\lambda\mapsto u_\lambda^*$.