Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

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 -