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.


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