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Singularly perturbed nonlinear elliptic problems on manifolds.
- Source :
-
Calculus of Variations & Partial Differential Equations . Dec2005, Vol. 24 Issue 4, p459-477. 19p. - Publication Year :
- 2005
-
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 $${\cal M}$$ 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 09442669
- Volume :
- 24
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Calculus of Variations & Partial Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 19234913
- Full Text :
- https://doi.org/10.1007/s00526-005-0339-4