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On the role of mean curvature in some singularly perturbed Neumann problems. (English) Zbl 0942.35058
In this paper, $\Omega$ is a smooth domain in $\bbfR^n$, which satisfies the following condition: there exists an open bounded subset $\Lambda$ of its boundary $\partial\Omega$ with a smooth boundary $\partial\Lambda$ and closed subsets $B,B_0$ of $\Lambda$, such that $B$ is connected and $B_0\subset B$. The authors consider the problem (1): $\varepsilon^2\Delta u-u+f(u)=0$ and $u>0$ in $\Omega$ with the boundary condition ${\partial u\over\partial r}=0$ on $\partial\Omega$. Here $f$ is a suitable function $\bbfR\to \bbfR$; the example $f(t)=t^p$, $p>1$ and $p < {n+2\over n-2}$ if $n\ge 3$, is allowed. Let $H(P)$ be the mean curvature function at $P\in\partial \Omega$. Let $c$ be a topologically nontrivial critical point for $H$ in $\Lambda$, characterized through a max-min scheme. The main result states the existence of a boundary-spike family of solutions of (1), with maxima $P_\varepsilon\in\Lambda$, so that $H(P_\varepsilon)\to c$. As in {\it M. del Pino} and {\it P. L. Felmer} [J. Funct. Anal. 149, 245-265 (1997; Zbl 0887.35058)], the proof is based on the introduction of a modified version of the energy-functional and ideas developed by W.-M. Ni, I. Takagi and J. Wei in recent papers on related problems.
Reviewer: D.Huet (Nancy)

35J20Second order elliptic equations, variational methods
35B40Asymptotic behavior of solutions of PDE
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