Perturbation Methods and Semilinear Elliptic Problems on R^n

By A. Ambrosetti, A. Malchiodi. 2005

364 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/71b8acee4e4e245a7e0a6bd0190070845b4b083a

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Citations in the corpus

Semiclassical states for weakly coupled nonlinear Schrödinger systems (2008, background): “As a related problem, a large amount of work (see [1, 4, 5, 13, 15, 20, 27] and references therein) has been devoted to the study of the semiclassical states for (NLS), namely the study of the singularly perturbed equation −ε21u + V (x)u = u3 in R3 for ε going to zero, whereV (x) is a potential mode”
On Schrödinger-Poisson Systems (2008, methodology): “which has been broadly investigated, see e.g. the monograph [ 1 ] and references therein.”
Multi-bump solitons to linearly coupled systems of nonlinear Schrödinger equations (2007, background): “Remark 2.5 The particular case in which (u) = 0(u) + ( u), and Z is a critical manifold of 0, namely 00(z) = 0 for every z 2 Z, has been broadly discussed in [ 6, Chapter 2 ]. Assumptions (A1) and (A3) are obviously satisfied, while (A2) follows from the assumption that Z is non-degenerate, in the s”
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum (2014, background): “when s = 1 and the operator boils down to the classical Laplacian), there is a broad literature on concentration phenomena: we recall here the seminal papers [28, 29] and we refer to [2] for detailed discussions and more precise references.”
Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin} (2012, positive): “(i i) We treat more general nonlinearity f than in [20,27,3,2,17,28,21].”
Semi-classical states for the Choquard equation (2013, background): “…Schrödinger equation which is the formal limit of the Choquard equation ( P ε ) as α → 0, such families of semi-classical solutions that concentrate, as ε → 0, around one or several critical points of the potential V have been constructed by various methods during the last decades [3,5,6,21,22,24].”
Multi-bump solutions of -Δn = K(x)u (n+2)/(n-2) on lattices in ℝ n (2018, methodology): “For subcritical exponent problems, we refer to Ambrosetti-Malchiodi [1], Gui-Wei [25], Malchiodi [39], Li-Nirenberg [37], Lin-Ni-Wei [38], Ao-WeiZeng [2] and the references therein.”