Singularly perturbed nonlinear elliptic problems on manifolds
SCIE
SCOPUS
- Title
- Singularly perturbed nonlinear elliptic problems on manifolds
- Authors
- Byeon, J; Park, J
- Date Issued
- 2005-12
- Publisher
- SPRINGER
- Abstract
- Let M be a connected compact smooth Riemannian manifold of dimension n >= 3 with or without smooth boundary aM. We consider the following singularly perturbed nonlinear elliptic problem on M epsilon(2)Delta(M)u - u + f(u) = 0, u > 0 on M, au/av = 0 on a M where Am is the Laplace-Beltrami operator on M, v is an exterior normal to aM and a nonlinearity f of subcritical growth. For certain f, there exists a mountain pass solution it, 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 aM = theta(aM not equal theta), the peak point x(epsilon) of the solution it, converges to a maximum point of the scalar curvature S on M(the mean curvature H on aM) as epsilon -> 0, respectively.
- Keywords
- LEAST-ENERGY SOLUTIONS; SEMILINEAR NEUMANN PROBLEM; POSITIVE SOLUTIONS; EQUATIONS; EXISTENCE; SYSTEM
- URI
- https://oasis.postech.ac.kr/handle/2014.oak/29543
- DOI
- 10.1007/S00526-005-0
- ISSN
- 0944-2669
- Article Type
- Article
- Citation
- CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, vol. 24, no. 4, page. 459 - 477, 2005-12
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