Robin double-phase problems with singular and superlinear terms

TL;DR

This paper studies a nonlinear Robin boundary value problem involving combined p- and q-Laplacian operators with singular and superlinear reaction terms, analyzing how the set of positive solutions changes with a parameter using variational methods.

Contribution

It introduces a bifurcation analysis for a Robin problem with combined p- and q-Laplacian operators and singular, superlinear reactions without the Ambrosetti-Rabinowitz condition.

Findings

Bifurcation-type results describing solution set changes with parameter .

Existence of positive solutions under complex nonlinear conditions.

Application of variational, truncation, and comparison techniques.

Abstract

We consider a nonlinear Robin problem driven by the sum of $p$-Laplacian and $q$-Laplacian (i.e. the $(p,q)$-equation). In the reaction there are competing effects of a singular term and a parametric perturbation $\lambda f(z,x)$, which is Carath\'eodory and $(p-1)$-superlinear at $x\in\mathbb{R},$ without satisfying the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $\lambda>0$ varies.