Multiple perturbations of a singular eigenvalue problem

TL;DR

This paper investigates how critical and superlinear subcritical perturbations affect solutions to a quasilinear elliptic equation with a singular potential, using variational methods and concentration-compactness principles.

Contribution

It introduces a novel analysis of perturbations in singular eigenvalue problems, establishing existence results under specific parameter conditions.

Findings

Existence of solutions for positive parameter values below the principal eigenvalue.

Application of concentration-compactness in the singular setting.

Extension of variational methods to singular quasilinear problems.

Abstract

We study the perturbation by a critical term and a $(p-1)$-superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential. By means of variational arguments and a version of the concentration-compactness principle in the singular case, we prove the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem.