Bifurcation and multiplicity results for critical $p$-Laplacian problems

Abstract

We prove a bifurcation and multiplicity result that is independent of the dimension $N$ for a critical $p$-Laplacian problem that is an analog of the Brezis-Nirenberg problem for the quasilinear case. This extends a result in the literature for the semilinear case $p = 2$ to all $p \in (1,\infty)$. In particular, it gives a new existence result when $N \lt p^2$. When $p \ne 2$ the nonlinear operator $- \Delta_p$ has no linear eigenspaces, so our extension is nontrivial and requires a new abstract critical point theorem that is not based on linear subspaces. We prove a new abstract result based on a pseudo-index related to the $\mathbb Z_2$-cohomological index that is applicable here.