A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL

doi:10.1007/s10473-022-0606-x

Abstract

Abstract: In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential i$?_tu+\Delta u+\frac{K}{|x|}u=\lambda|u|^{p-1}u$ with 1<p≤5 on $\mathbb{R}^3$. Our results reveal the influence of the long range potential $K|x|^{-1}$ on the existence and scattering theories for nonlinear Schrödinger equations. In particular, we prove the global existence when the Coulomb potential is attractive, i.e., when $K>0$, and the scattering theory when the Coulomb potential is repulsive, i.e., when $K\leq0$. The argument is based on the newly-established interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.

Key words: nonlinear Schrödinger equations, long range potential, global well-posedness, blow-up, scattering

CLC Number:

35P25

Changxing MIAO, Junyong ZHANG, Jiqiang ZHENG. A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL[J].Acta mathematica scientia,Series B, 2022, 42(6): 2230-2256.

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