Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Antonio Ambrosetti; Veronica Felli; Andrea Malchiodi

Journal of the European Mathematical Society (2005)

  • Volume: 007, Issue: 1, page 117-144
  • ISSN: 1435-9855

Abstract

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We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V ( x ) | x | α , 0 < α < 2 , and K ( x ) | x | β , β > 0 . Working in weighted Sobolev spaces, the existence of ground states v ε belonging to W 1 , 2 ( N ) is proved under the assumption that σ < p < ( N + 2 ) / ( N 2 ) for some σ = σ N , α , β . Furthermore, it is shown that v ε are spikes concentrating at a minimum point of 𝒜 = V θ K 2 / ( p 1 ) , where θ = ( p + 1 ) / ( p 1 ) 1 / 2 .

How to cite

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Ambrosetti, Antonio, Felli, Veronica, and Malchiodi, Andrea. "Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity." Journal of the European Mathematical Society 007.1 (2005): 117-144. <http://eudml.org/doc/277564>.

@article{Ambrosetti2005,
abstract = {We deal with a class on nonlinear Schrödinger equations (NLS) 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 $\theta =(p+1)/(p−1)−1/2$.},
author = {Ambrosetti, Antonio, Felli, Veronica, Malchiodi, Andrea},
journal = {Journal of the European Mathematical Society},
keywords = {nonlinear Schrödinger equations; weighted Sobolev spaces; critical point; standing wave; existence of ground states; critical point; standing wave; nonlinear Schrödinger equations; weighted Sobolev spaces; existence of ground states},
language = {eng},
number = {1},
pages = {117-144},
publisher = {European Mathematical Society Publishing House},
title = {Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity},
url = {http://eudml.org/doc/277564},
volume = {007},
year = {2005},
}

TY - JOUR
AU - Ambrosetti, Antonio
AU - Felli, Veronica
AU - Malchiodi, Andrea
TI - Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity
JO - Journal of the European Mathematical Society
PY - 2005
PB - European Mathematical Society Publishing House
VL - 007
IS - 1
SP - 117
EP - 144
AB - We deal with a class on nonlinear Schrödinger equations (NLS) 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 $\theta =(p+1)/(p−1)−1/2$.
LA - eng
KW - nonlinear Schrödinger equations; weighted Sobolev spaces; critical point; standing wave; existence of ground states; critical point; standing wave; nonlinear Schrödinger equations; weighted Sobolev spaces; existence of ground states
UR - http://eudml.org/doc/277564
ER -

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