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

Jaeyoung ByeonKazunaga Tanaka — 2013

Journal of the European Mathematical Society

We consider a singularly perturbed elliptic equation ϵ 2 Δ u - V ( x ) u + f ( u ) = 0 , u ( x ) > 0 on N , 𝚕𝚒𝚖 x u ( x ) = 0 , where V ( x ) > 0 for any x N . The singularly perturbed problem has corresponding limiting problems Δ U - c U + f ( U ) = 0 , U ( x ) > 0 on N , 𝚕𝚒𝚖 x U ( x ) = 0 , c > 0 . Berestycki-Lions found almost necessary and sufficient conditions on 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 potential V under possibly general conditions on f . In...

Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials

Jaeyoung ByeonZhi-Qiang Wang — 2006

Journal of the European Mathematical Society

For singularly perturbed Schrödinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti–Malchiodi–Ni [3] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in [3].

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