Journal of the European Mathematical Society


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Volume 7, Issue 1, 2005, pp. 117–144
DOI: 10.4171/JEMS/24

Published online: 2005-03-31

Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Antonio Ambrosetti[1], Veronica Felli[2] and Andrea Malchiodi[3]

(1) SISSA, Trieste, Italy
(2) Università degli Studi di Milano-Bicocca, Italy
(3) Scuola Normale Superiore, Pisa, Italy

We deal with a class on nonlinear Schr\"odinger equations \eqref{eq:1} with potentials $V(x)\sim |x|^{-\a}$, $0<\a<2$, and $K(x)\sim |x|^{-\b}$, $\b>0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\e}$ belonging to $W^{1,2}(\Rn)$ is proved under the assumption that $p$ satisfies \eqref{eq:p}. Furthermore, it is shown that $v_{\e}$ are {\em spikes} concentrating at a minimum of ${\cal A}=V^{\theta}K^{-2/(p-1)}$, where $\theta= (p+1)/(p-1)-1/2$.

Keywords: Nonlinear Schrödinger equations, weighted Sobolev spaces

Ambrosetti Antonio, Felli Veronica, Malchiodi Andrea: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. 7 (2005), 117-144. doi: 10.4171/JEMS/24