JournalsjemsVol. 7, No. 1pp. 117–144

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
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity cover
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Abstract

We deal with a class on nonlinear Schr\"odinger equations \eqref{eq:1} with potentials V(x)x\aV(x)\sim |x|^{-\a}, 0<\a<20<\a<2, and K(x)x\bK(x)\sim |x|^{-\b}, \b>0\b>0. Working in weighted Sobolev spaces, the existence of ground states v\ev_{\e} belonging to W1,2(\Rn)W^{1,2}(\Rn) is proved under the assumption that pp satisfies \eqref{eq

}. Furthermore, it is shown that v\ev_{\e} are {\em spikes} concentrating at a minimum of A=VθK2/(p1){\cal A}=V^{\theta}K^{-2/(p-1)}, where θ=(p+1)/(p1)1/2\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