N-Laplacian problems with critical Trudinger-Moser nonlinearities

Abstract

We prove existence and multiplicity results for a $N$-Laplacian problem with a critical exponential nonlinearity that is a natural analog of the Brezis-Nirenberg problem for the borderline case of the Sobolev inequality. This extends results in the literature for the semilinear case $N = 2$ to all $N \ge 2$. When $N > 2$ the nonlinear operator $- \Delta_N$ has no linear eigenspaces and hence this extension requires new abstract critical point theorems that are not based on linear subspaces. We prove new abstract results based on the ${\mathbb Z}_2$-cohomological index and a related pseudo-index that are applicable here.