Characterization of concentration points and L∞-estimates for solutions of a semilinear neumann problem involving the critical sobolev exponent

Abstract

$\Omega\subset\mathbb{R}^n(n\geq7)$ be a bounded domain with smooth boundary. For $\lambda>0$, let $u_\lambda$ be a solution of $$ \begin{align} -\Delta u+\lambda u&=u^{n+2\over n-2}\quad\rm{in }\quad\Omega, u&>0\quad\rm{in }\quad\Omega, {\partial u\over\partial\nu}&=0\quad\rm{on }\quad\partial\Omega, \end{align} $$ whose energy is less than the first critical level. Here we study the blow up points and the $L^\infty$-estimates of $u_\lambda$ as $\lambda\to\infty$. We show that the blow up points are the critical points of the mean curvature on the boundary.