Asymmetric critical $p$-Laplacian problems

Abstract

We obtain nontrivial solutions for two types of critical $p$-Laplacian problems with asymmetric nonlinearities in a smooth bounded domain in ${\mathbb{R}}^N,N\ge 2$. For $p<N$, we consider an asymmetric problem involving the critical Sobolev exponent $p^{\ast }=Np/(N-p)$. In the borderline case $p=N$, we consider an asymmetric critical exponential nonlinearity of the Trudinger-Moser type. In the absence of a suitable direct sum decomposition, we use a linking theorem based on the ${\mathbb{Z}}_2$-cohomological index to obtain our solutions.

arXiv admin note: text overlap with arXiv:1406.6242, arXiv:1411.2198