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Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity. (English) Zbl 0793.35033
Summary: We consider the problem: \(-\Delta u+\lambda u=u^{(n+2)/(n-2)}\), \(u>0\) in \(\Omega\), \(\partial u/ \partial \nu=0\) on \(\partial \Omega\), where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^ n\) \((n \geq 3)\). We show that, for \(\lambda\) large, least-energy solutions of the above problem have a unique maximum point \(P_ \lambda\) on \(\partial \Omega\) and the limit points of \(P_ \lambda\), as \(\lambda \to \infty\) are contained in the set of the points of maximum mean curvature. We also prove that, if \(\partial \Omega\) has \(k\) peaks then the equation has at least \(k\) solutions for \(\lambda\) large.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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