# Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

### Antonio Ambrosetti

SISSA, Trieste, Italy### Andrea Malchiodi

Scuola Normale Superiore, Pisa, Italy### Veronica Felli

Università degli Studi di Milano-Bicocca, Italy

## Abstract

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

}. Furthermore, it is shown that $v_{\e}$ are {\em spikes} concentrating at a minimum of ${\cal A}=V^{\theta}K^{-2/(p-1)}$, where $\theta= (p+1)/(p-1)-1/2$.## Cite this article

Antonio Ambrosetti, Andrea Malchiodi, Veronica Felli, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. 7 (2005), no. 1, pp. 117–144

DOI 10.4171/JEMS/24