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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 $$\lim_{n\to\infty} \|\nabla u_ \lambda- \nabla U_{\varepsilon_ \lambda, P_ \lambda} \|_{L^ 2 (\Omega)} =0$$.

MSC:
 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
Keywords:
critical Sobolev exponent