Standing waves for supercritical nonlinear Schrödinger equations

@inproceedings{Dvila2007StandingWF,
  title={Standing waves for supercritical nonlinear Schr{\"o}dinger equations},
  author={Juan D{\'a}vila and Manuel del Pino and Monica Musso and Juncheng Wei},
  year={2007}
}
Let V(x) be a non-negative, bounded potential in RN, N⩾3 and p supercritical, p>N+2N−2. We look for positive solutions of the standing-wave nonlinear Schrodinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(|x|−2) as |x|→+∞, then for N⩾4 and p>N+1N−3 this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|−μ) with μ>N, then this result still holds provided that N⩾3 and p>N+2N−2. Other conditions for solvability, involving… CONTINUE READING

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