Asymmetric critical $p$-Laplacian problems
  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. 
Kanishka Perera, Yang Yang, and Zhitao Zhang
math.AP