JournalsjemsVol. 15, No. 5pp. 1859–1899

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

  • Jaeyoung Byeon

    KAIST, Daejeon, South Korea
  • Kazunaga Tanaka

    Waseda University, Tokyo, Japan
Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential cover
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Abstract

We consider a singularly perturbed elliptic equation

\e2ΔuV(x)u+f(u)=0, u(x)>0 on \RN,limxu(x)=0,\e^2\Delta u - V(x) u + f(u)=0, \ u(x) > 0 \textrm{ on } \RN, \, \lim_{|x| \to \infty}u(x) = 0,

where V(x)>0V(x) > 0 for any x\RN.x \in \RN. The singularly perturbed problem has corresponding limiting problems

ΔUcU+f(U)=0, U(x)>0 on  \RN,limxU(x)=0, c>0.\Delta U - c U + f(U)=0, \ U(x) > 0 \textrm{ on } \ \RN, \, \lim_{|x| \to \infty}U(x) = 0, \ c > 0.

Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity ff 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 potential VV under possibly general conditions on ff. In this paper, we prove that under the optimal conditions of Berestycki-Lions on fC1f \in C^1, there exists a solution concentrating around topologically stable positive critical points of VV, whose critical values are characterized by minimax methods.

Cite this article

Jaeyoung Byeon, Kazunaga Tanaka, Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential. J. Eur. Math. Soc. 15 (2013), no. 5, pp. 1859–1899

DOI 10.4171/JEMS/407