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Singularly perturbed nonlinear elliptic problems on manifolds

Authors:

Jaeyoung Byeon Junsang Park

Institution:

(1) Department of Mathematics, POSTECH, Pohang, Kyungbuk, 790-784, Republic of Korea

Abstract:

Let ${\cal M}$ be a connected compact smooth Riemannian manifold of dimension $n \ge 3$ with or without smooth boundary $\partial {\cal M}.$ We consider the following singularly perturbed nonlinear elliptic problem on ${\cal M}$

$\varepsilon^2 \Delta_{{\cal M}} u - u + f(u)=0, \ \ u > 0 \quad {\rm on} \quad {\cal M}, \quad \frac{\partial u}{\partial \nu}=0 {\rm on } \partial {\cal M}$

where $\Delta_{{\cal M}}$ is the Laplace-Beltrami operator on ${\cal M}$ , $\nu$ is an exterior normal to $\partial {\cal M}$ and a nonlinearity $$f$$

of subcritical growth. For certain $$f,$$

there exists a mountain pass solution $u_\varepsilon$ of above problem which exhibits a spike layer. We are interested in the asymptotic behaviour of the spike layer. Without any non-degeneracy condition and monotonicity of $$f(t)/t,$$

we show that if $\partial {\cal M} =\emptyset(\partial {\cal M} \ne \emptyset),$ the peak point $x_\varepsilon$ of the solution $u_\varepsilon$ converges to a maximum point of the scalar curvature $$S$$

on ${\cal M}$ (the mean curvature $$H$$

on $\partial {\cal M})$ as $\varepsilon \to 0,$ respectively. The research of the first author was supported in part by KRF-2002-070-C000005 of Korea Research Foundation.

Keywords:

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