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 \(\mathbb{R}^n\), 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 \(\mathbb{R}\to \mathbb{R}\); the example \(f(t)=t^p\), \(p>1\) and \(p < {n+2\over n-2}\) if \(n\geq 3\), 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 M. del Pino and 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)


35J20 Variational methods for second-order elliptic equations
35B40 Asymptotic behavior of solutions to PDEs


Zbl 0887.35058
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