On some nonlinear partial differential equations involving the $p$ ‐Laplacian and critical Sobolev trace maps

In this article, we study some partial differential equations with Neumann nonlinear boundary conditions, related to the problem of the best constant from $W^{1,p}(\varOmega)$ to $L^{p^\star}(\curpartial\varOmega)$ , where $p^\star$ is the critical Sobolev exponent for the embedding of $W^{1,p}(\varOmega)$ into $L^{q}(\curpartial\varOmega)$ ( $\varOmega $ is a bounded $\mathcal{C}^1$ open set and $p<N$ ).