On interacting bumps of semi-classical states of nonlinear Schrödinger equations

By Xiaosong Kang, Jun-cheng Wei. Advances in Differential Equations, 2000

187 citations recorded by the source corpus.

We study concentrated positive bound states of the following nonlinear Schrr odinger equation h 2 u ? V (x)u + u p = 0; u > 0; x 2 R N where p is subcritical. We prove that, at a local maximum point x 0 of the potential function V (x) and for arbitrary positive integer K (K > 1), there always exist solutions with K interacting bumps concentrating near x 0. We also prove that at a nondegenerate local minimum point of V (x) such solutions do not exist.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dabc982f6c7fd098d1c8bd7a9ea8b1698779e4d2

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Citations in the corpus

Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations (2015, background): “For other results concerning the existence of the solutions with the concentration phenomena, one can refer to [2, 3, 5, 11, 14, 16, 19, 23, 24, 26] and the references therein.”
The toda system and multiple-end solutions of autonomous planar elliptic problems (2007, methodology): “We refer the reader to the works [2, 3, 7, 11, 12, 13, 16, 20, 21, 27, 29, 30, 39, 37, 38] and references therein.”
Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials (2007, background): “For concentrations on higher-dimensional sets, we refer to Ambrosetti, Malchiodi and Ni [2], and M. del Pino, M. Kowalczyk and Wei [13].”
Singularly Perturbed Elliptic Equations with Symmetry: Existence of Solutions Concentrating on Spheres, Part I (2003, background): “We remark that, if p is subcritical, then near local maxima of V there exist multi-spike solutions, see [23].”
Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: One and Two Dimensional Cases (2008, background): “Motivated by the approach in [19], many authors have obtained refined results on (3) in higher dimension and for more general f (see [2, 13, 14, 25, 26, 28, 29]).”
Concentrating standing waves for the fractional nonlinear Schr\"odinger equation (2013, methodology): “[18] X. Kang, J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations.”
Infinitely many positive solutions for nonlinear equations with non-symmetric potentials (2012, background): “4) where ε > 0 is a small parameter and p subcritical, then the number of the critical points of V (y) (see for example [1, 36],[17]–[20],[23, 36, 45]), the type of the critical points of V (y) (see for example [8, 26, 35], and the topology of the level set of V (y) [2, 3, 9, 24], can affect the num”
Infinitely many positive solutions for the nonlinear Shcrodinger equations in $R^N$ (2008, background): “6) where ε > 0 is a small parameter, then the number of the critical points of V (y) (or Q(y)) (see for example [1, 7, 8],[13]–[16],[18, 27, 29]), the type of the critical points of V (y) (or Q(y)) (see for example [12, 21, 26], and the topology of the level set of V (y) (or Q(y)) [2, 3, 11, 19], ca”
Spikes in Two-component Systems of Nonlinear Schrodinger Equations with Trapping Potentials (2006, background): “We refer the reader for instance to [1, 3, 2, 4, 5, 8, 9, 10, 12, 14, 15, 16, 17, 18, 22, 23, 25, 26] and the references therein.”
Strongly interacting bumps for the Schrödinger–Newton equations (2009, methodology): “In the same way as in [16] it can be shown that for h small the optimization problem”
Solutions of perturbed Schrödinger equations with critical nonlinearity (2007, background): “See for example [1,9,10,12–21,23] and the references therein.”
Existence and Concentration Result for the Kirchhoff Type Equations with General Nonlinearities (2014, unknown): “For instance, we refer to [5–7,20,21,25– 27,33,44,48,54,55].”
Semi-classical standing waves for nonlinear Schrödinger equations at structurally stable critical points of the potential (2013, result): “Motivated by the approach in [20], many authors have obtained further refined results in higher dimensions for more general f and more general types of critical points of V (see [1, 2, 19, 27, 28, 32, 33] and references therein).”
The Toda System and Clustering Interfaces in the Allen–Cahn equation (2008, background): “In the one-dimensional or radial cases, concentration in the form of clusters of transition layers or spikes has already been observed in various problems in the literature, see [4,10,11,13,27,28,32,33], phenomena in accordance with higherdimensional multi-spike clustering as predicted in [20].”
Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations (2007, background): “Moreover, semiclassical states with several peaks near a point of maximum of V , have been found in [13].”