On the location of the peaks of least-energy solutions to semilinear Dirichlet problems with critical growth

Abstract

We study the location of the peaks of solution for the critical growth problem $-{\epsilon }^2\Delta u+u=f\left(u\right)+u^{2^{*}-1}$, $u>0$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a bounded domain; $2^{*}=2N/\left(N-2\right)$, $N\ge 3$, is the critical Sobolev exponent and $f$ has a behavior like $u^p$, .