## Differential and Integral Equations

### Characterization of concentration points and $L^\infty$-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.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 1 (1995), 41-68.

Dates
First available in Project Euclid: 21 May 2013

Adimurthi; Pacella, Filomena; Yadava, S. L. Characterization of concentration points and $L^\infty$-estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent. Differential Integral Equations 8 (1995), no. 1, 41--68. https://projecteuclid.org/euclid.die/1369143783