Singularly Perturbed Elliptic Equations with Symmetry: Existence of Solutions Concentrating on Spheres, Part I

By A. Ambrosetti, A. Malchiodi, Wei-Ming Ni. 2003

231 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/5e19941a6a4a72df5ce37d87b2e4fd9d509f5c9f

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Some new entire solutions of semilinear elliptic equations on Rn (2009, background): “Existence of such solutions was shown in [ 3 ], [4], [6], [9] under some symmetry assumptions, and then in [18] the authors proved concentration along curves for n = 2 (see also [38], [39] for general n), in a non-symmetric setting.”
SEMICLASSICAL STATES FOR COUPLED SCHRÖDINGER–MAXWELL EQUATIONS: CONCENTRATION AROUND A SPHERE (2005, methodology): “We point out that our problem turns out to be qualitatively dierent from [ 5 ].”
Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations (2007, background): “[6, 23] which contain several further references), concentration and multi-bump phenomena for semiclassical states (see e.g. [ 7 , 8, 10, 12, 15, 19]), to cite a few.”
Nonlinear schrodinger equations with unbounded and decaying radial potentials (2007, unknown): “for all p > 1 as long as † is su‐ciently small. Note that the solution given in [ 2 ]”
On Schrödinger-Poisson Systems (2008, result): “In particular, if K ≡ 0, we recover the result proved in [ 2 ].”
Elliptic equations with critical exponent on spherical caps of S3 (2006, background): “When V is a radial function, the result of [1], [2], combined with the Pohozaev inequality, establishes”
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.”
Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains (2005, background): “Other results regarding concentration at higher dimensional sets are given in [ 3 ], [4], [5], [6], [9], [30] under some symmetry assumptions.”
Concentration on curves for nonlinear Schrödinger Equations (2007, background): “In particularthisestablishesthevalidity of a conjectureraisedin [3] in thetwodimensional case.”
Spherical semiclassical states of a critical frequency for Schrodinger equations with decaying potentials (2006, methodology): “The results generalize some recent work of Ambrosetti–Malchiodi–Ni [3] which gives solutions concentrating on spheres where the potential is positive.”