## Differential and Integral Equations

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

### Characterization of concentration points and $L^\infty$-estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent

Adimurthi, Filomena Pacella, and S. L. Yadava

#### 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

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1369143783

**Mathematical Reviews number (MathSciNet)**

MR1296109

**Zentralblatt MATH identifier**

0814.35029

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations

Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

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