A bifurcation and multiplicity result for a critical growth elliptic problem

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