Semiclassical States of Nonlinear Schrödinger Equations

Ambrosetti, Antonio; Badiale, Marino; Cingolani, Silvia

doi:10.1007/s002050050067

Cited by 460 publications

(474 citation statements)

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“…We refer, e.g., to [10] for HartreeFock equations, [5] for Schrödinger operators of atoms and molecules, and [1], [3], [4], [6], [8], [12], [14] for the Schödinger operators with continuous (periodic etc.) potentials.…”

Section: Recall That the Operatormentioning

confidence: 99%

Solutions to a class of Schrödinger equations

Ding

2001

Proc. Amer. Math. Soc.

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Section: Recall That the Operatormentioning

confidence: 99%

Solutions to a class of Schrödinger equations

Ding

2001

Proc. Amer. Math. Soc.

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“…Their method, based on an interesting Lyapunov-Schmidt finite dimensional reduction, was extended by Oh in [28,29] to include a similar result in higher dimensions, provided 1 < p < N +2 N −2 . Other existence results for positive solutions of problem (1.2) under the condition inf x∈R N W (x) > E can be found in [1,2,3,10,12,13,14,15,16,17,20,22,24,25,27,31,32].…”

Section: Introductionmentioning

confidence: 98%

Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

Cao

Noussair

Yan

2008

Trans. Amer. Math. Soc.

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Abstract. In this paper we study the existence and qualitative property of standing wave solutions ψ(x, t) = e ∆ψ − W (x)ψ + |ψ| p−1 ψ = 0 with E being a critical frequency in the sense that infsuch that the interior of Z i is not empty and ∂Z i is smooth, V has t isolated zero points, b i , i = 1, · · · , t, and V has l critical points a i (i = 1, · · · , l) such that V (a i ) > 0, then for > 0 small, there exists a standing wave solution which is trapped in a neighborhoodMoreover the amplitudes of the standing wave aroundare of a different order of . This type of multi-scale solution has never before been obtained.

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“…Related results were proved e.g. in [12], [18], [9] (where a "local" version of Ni-Wei's result is proved) and in [2], [6], [11] (where a linear term V (x)u is added to equation (1.5) while Ω = R N ). The subject was revisited by Del Pino and Felmer, for both Neumann and Dirichlet boundary conditions, in [5], where shorter and more elementay arguments were introduced, with respect to those in [13,14].…”

Section: Introductionmentioning

confidence: 98%

Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions

Pistoia

Ramos

2008

Nonlinear differ. equ. appl.

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Abstract. We consider a system of the form −ε 2 ∆u + u = g(v), −ε 2 ∆v + v = f (u) in Ω with Dirichlet boundary condition on ∂Ω, where Ω is a smooth bounded domain in R N , N 3 and f, g are power-type nonlinearities having superlinear and subcritical growth at infinity. We prove that the least energy solutions to such a system concentrate, as ε goes to zero, at a point of Ω which maximizes the distance to the boundary of Ω.2000 Mathematics Subject Classification: 35J50, 35J55, 38E05.

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Semiclassical States of Nonlinear Schrödinger Equations

Cited by 460 publications

References 9 publications

Solutions to a class of Schrödinger equations

Solutions to a class of Schrödinger equations

Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations

Locating the peaks of the least energy solutions to an elliptic system with Dirichlet boundary conditions

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