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

We consider a critical problem in a bounded domain involving the $p$-Grushin operator ${\mathrm{\Delta }}_{\alpha }^p$. After a truncation argument, we obtain infinitely many solutions to our problem via Krasnoselskii’s genus, extending a previous result of García 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.