## Characterization of concentration points and $$L^ \infty$$-estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent.(English)Zbl 0814.35029

Summary: Let $$\Omega \subset \mathbb{R}^ n$$ $$(n \geq 7)$$ be a bounded domain with smooth boundary. For $$\lambda > 0$$, let $$u_ \lambda$$ be a solution of $-\Delta u + \lambda u = u^{(n+2)/(n-2)},\;u > 0 \quad \text{in} \quad \Omega,\;\partial u/ \partial \nu = 0 \quad \text{on} \quad \partial \Omega$ whose energy is less than the first critical level. 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.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B40 Asymptotic behavior of solutions to PDEs 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs