A bifurcation and multiplicity result for a critical growth elliptic problem

We consider a Brézis–Nirenberg type critical growth $p$-Laplacian problem involving a parameter $\mu >0$ in a smooth bounded domain $\Omega$. We prove the existence of multiple nontrivial solutions if either $\mu$ or the volume of $\Omega$ is sufficiently small. The proof is based on an abstract critical point theorem that only assumes a local ${\left(\text{PS}\right)}_{}$ condition. Our results are new even in the semilinear case $p=2$.