Semi-classical standing waves for nonlinear Schrodinger equations at structurally stable critical points of the potential

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We consider a singularly perturbed elliptic equation epsilon(2)Delta u - V(x)u + f(u) = 0, u(x) > 0 on R-N, lim(vertical bar x vertical bar ->infinity) u(x) = 0, where V(x) > 0 for any x is an element of R-N. The singularly perturbed problem has corresponding limiting problems Delta U - cU + f(U) = 0, U(x) > 0 on R-N, lim(vertical bar x vertical bar ->infinity) u(x) = 0, c > 0. Berestycki-Lions [3] found almost necessary and sufficient conditions on the nonlinearity f for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of the potential V under possibly general conditions on f. In this paper, we prove that under the optimal conditions of Berestycki-Lions on f is an element of C-1, there exists a solution concentrating around topologically stable positive critical points of V, whose critical values are characterized by minimax methods.
Publisher
EUROPEAN MATHEMATICAL SOC
Issue Date
2013
Language
English
Article Type
Article
Citation

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, v.15, no.5, pp.1859 - 1899

ISSN
1435-9855
DOI
10.4171/JEMS/407
URI
http://hdl.handle.net/10203/255309
Appears in Collection
MA-Journal Papers(저널논문)
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