Bound state solutions for a class of nonlinear Schrödinger equations
type: Item
instance of: scholarly article
title: Bound state solutions for a class of nonlinear Schrödinger equations
zbMATH Open document ID: 1156.35084
DOI: 10.4171/RMI/537
author: Denis Bonheure
author: Jean Van Schaftingen
published in: Revista Matemática Iberoamericana
publication date: 2008-08-06T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmi/1216247103
full work available at URL: https://eudml.org/doc/42374
review text: Summary: We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form \[ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, \] where \(V, K\) are positive continuous functions and \(p > 1\) is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential \(V\) is allowed to vanish at infinity and the competing function \(K\) does not have to be bounded. In the semi-classical limit, i.e. for \(\varepsilon\sim 0\), we prove the existence of bound state solutions localized around local minimum points of the auxiliary function \(\mathcal{A} = V^\theta K^{-\frac{2}{p-1}}\), where \(\theta=(p+1)/(p-1)-N/2\). A special attention is devoted to the qualitative properties of these solutions as \(\varepsilon\) goes to zero.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 5307135
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: semi-classical states
zbMATH Keywords: concentration
zbMATH Keywords: vanishing potentials
zbMATH Keywords: unbounded competition functions
MaRDI profile type: Publication
cites work: Semiclassical states of nonlinear Schrödinger equations
cites work: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity
cites work: Q3057556
cites work: Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I
cites work: Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, Part II
cites work: Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity
cites work: Multiplicity results for some nonlinear Schrödinger equations with potentials
cites work: Radial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials
cites work: On the existence of a positive solution of semilinear elliptic equations in unbounded domains
cites work: Nonlinear Schrödinger equations with potentials vanishing at infinity
cites work: Standing waves with a critical frequency for nonlinear Schrödinger equations
cites work: Standing waves with a critical frequency for nonlinear Schrödinger equations. II.
cites work: Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials
cites work: Local mountain passes for semilinear elliptic problems in unbounded domains
cites work: Semi-classical states for nonlinear Schrödinger equations
cites work: Multi-peak bound states for nonlinear Schrödinger equations
cites work: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method
cites work: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential
cites work: Q4942054
cites work: Q3923023
cites work: Q3470840
cites work: Q3683006
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. II
cites work: Remarks on a semilinear elliptic equation on \({\mathbb{R}}^ n\)
cites work: On a singularly perturbed elliptic equation
cites work: Existence of Semiclassical Bound States of Nonlinear Schrödinger Equations with Potentials of the Class (V)a
cites work: Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials
cites work: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential
cites work: Q3997067
cites work: Multi-peak solutions for a class of nonlinear Schrödinger equations
cites work: On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results
cites work: Q3749777
cites work: On a class of nonlinear Schrödinger equations
cites work: Stability for semilinear parabolic equations with decaying potentials in \(\mathbb{R}^n\) and dynamical approach to the existence of ground states
cites work: NONLINEAR SCHRÖDINGER EQUATIONS WITH UNBOUNDED AND DECAYING RADIAL POTENTIALS
cites work: Symmetry of solutions of semilinear elliptic problems
cites work: On concentration of positive bound states of nonlinear Schrödinger equations
cites work: On Concentration of Positive Bound States of Nonlinear Schrödinger Equations with Competing Potential Functions
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description: scientific article; zbMATH DE number 5307135