Singularly perturbed nonlinear elliptic problems on manifolds

Jaeyoung Byeon; Junsang Park
December 2005
Calculus of Variations & Partial Differential Equations;Dec2005, Vol. 24 Issue 4, p459
Academic Journal
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 $${\cal M}$$ 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.


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