A note on critical problems involving the $p$-Grushin Operator: existence of infinitely many solutions

TL;DR

This paper establishes the existence of infinitely many solutions for a critical boundary value problem involving the $p$-Grushin operator, extending previous results to a more general operator using variational methods.

Contribution

It extends prior work by proving the existence of infinitely many solutions for the $p$-Grushin operator case using Krasnoselskii's genus and a truncation technique.

Findings

Successfully verified the Palais-Smale condition under a certain level.

Extended the existence results to the $p$-Grushin operator.

Demonstrated the applicability of variational methods to this class of problems.

Abstract

We consider a critical problem in a bounded domain involving the $p$-Grushin operator $\Delta_\alpha^p$. After a truncation argument, we obtain infinitely many solutions to our problem via Krasnoselskii's genus, extending a previous result of Garc\'ia Azorero and Peral Alonso to the $p$-Grushin operator. A central part of our analysis is the verification of the Palais-Smale condition of the associated functional under a certain level.