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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
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