Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

• Autores: Veronica Felli, Antonio Ambrosetti, Andrea Malchiodi
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 7, Nº 1, 2005, págs. 117-144
• Idioma: inglés
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• Resumen

  • 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 $\theta= (p+1)/(p-1)-1/2$.