$N$-Laplacian problems with critical Trudinger-Moser nonlinearities

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 $-{\mathrm{\Delta }}_N$ 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.