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Remarks on a nonlinear Neumann problem with critical exponent. (English) Zbl 0817.35030
From the introduction: We give extensions of some results in [Adimurthi, F. Pacella and S. L. Yadava, J. Funct. Anal. 113, No. 2, 318-350 (1993; Zbl 0793.35033)] on the existence and qualitative behavior of solutions for the following nonlinear Neumann problem \[ (I)_ \lambda \qquad \qquad -\Delta u+ \lambda u= u^ p,\;\;u>0 \quad \text{in } \Omega, \qquad \partial u/\partial\nu =0 \quad \text{on } \partial\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^ N\) \((N\geq 3)\) with smooth boundary, \(\lambda>0\) is a constant and \(\nu\) is the unit outer normal to \(\partial\Omega\). We assume \(p= (N+2)/ (N-2)\) is the critical Sobolev exponent.
Theorem 1. Let \(N\geq 5\). Let \(u_ \lambda\) be a least energy solution of \((I)_ \lambda\) and \(P_ \lambda\in \partial\Omega\) be such that \(u_ \lambda (P_ \lambda)= \max_{x\in \overline {\Omega}} u_ \lambda (x)\). Then the limit points of \(P_ \lambda\), as \(\lambda\to \infty\), are contained in the set of the points of maximum mean curvature.
Theorem 2. Let \(N\geq 5\) and \(O_ 0\) be a strict local maximum point of \(H(x)\), such that \(H(P_ 0) >0\). Then there exists \(\lambda_ 0\) such that for \(\lambda\geq \lambda_ 0\), \((I)_ \lambda\) has a low energy solution \(u_ \lambda\) which attains its maximum over \(\overline {\Omega}\) at only one point \(P_ \lambda\in \partial \Omega\) with \(P_ \lambda\to P_ 0\), as \(\lambda\to \infty\). Moreover, setting \(\varepsilon_ \lambda= [u_ \lambda (P_ \lambda) ]^{-(p-1)/2}\), we have \(\lim_{n\to\infty} \|\nabla u_ \lambda- \nabla U_{\varepsilon_ \lambda, P_ \lambda} \|_{L^ 2 (\Omega)} =0\).

35J65 Nonlinear boundary value problems for linear elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35J20 Variational methods for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs