Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity
type: Item
instance of: scholarly article
title: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity
zbMATH Open document ID: 1064.35175
DOI: 10.4171/JEMS/24
author: A. Ambrosetti
author: Veronica Felli
author: Andrea Malchiodi
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2005-03-21T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: Summary: The authors deal with a class on nonlinear Schrödinger equations \[ -\varepsilon^2\Delta v+V(x)v=K(x)v^p,\quad x\in \mathbb{R}^N, \] with potentials \(V(x) \sim|x|^{-\alpha}\), \(0<\alpha<2\), and \(K(x)\sim|x|^{-\beta}\), \(\beta>0\). Working in weighted Sobolev spaces, the existence of ground states \(v_\varepsilon\) belonging to \(W^{1,2}(\mathbb{R}^N)\) is proved under the assumption that \(\sigma<p <(N+2)/(N-2)\) for some \(\sigma=\sigma_{N,\alpha,\beta}\). Furthermore, it is shown that \(v_\varepsilon\) are spikes concentrating at a minimum point of \({\mathcal A}=V^\theta K^{-2/(p-1)}\), where \(\theta= (p+1)/(p-1)-1/2\).
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 2148279
zbMATH Keywords: critical point
zbMATH Keywords: standing wave
zbMATH Keywords: nonlinear Schrödinger equations
zbMATH Keywords: weighted Sobolev spaces
zbMATH Keywords: existence of ground states
MaRDI profile type: Publication
cites work: Homoclinics: Poincaré-Melnikov type results via a variational approach
cites work: Variational perturbative methods and bifurcation of bound states from the essential spectrum
cites work: Semiclassical states of nonlinear Schrödinger equations
cites work: Remarks on a class of semilinear elliptic equations on \(\mathbb{R}^n\), via perturbation methods
OpenAlex ID: W2007460600
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description: scientific article; zbMATH DE number 2148279