Standing waves for supercritical nonlinear Schrödinger equations

@article{Dvila2007StandingWF,
title={Standing waves for supercritical nonlinear Schr{\"o}dinger equations},
author={J. D{\'a}vila and M. A. Pino and M. Musso and J. Wei},
journal={Journal of Differential Equations},
year={2007},
volume={236},
pages={164-198}
}

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