  In this paper we find estimates for the optimal constant in the critical
Sobolev trace inequality $S\|u\|^p_{L^{p_*}(\partial\Omega) \hookrightarrow
\|u\|^p_{W^{1,p}(\Omega)}$ that are independent of $\Omega$. This estimates
generalized those of [3] for general $p$. Here $p_* := p(N-1)/(N-p)$ is the
critical exponent for the immersion and $N$ is the space dimension. Then we
apply our results first to prove existence of positive solutions to a nonlinear
elliptic problem with a nonlinear boundary condition with critical growth on
the boundary, generalizing the results of [16]. Finally, we study an optimal
design problem with critical exponent.
