# 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

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topAmbrosetti, 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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