BOUND STATES OF SCHRÖDINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND VANISHING POTENTIALS

doi:10.1016/S0252-9602(17)30011-5

Abstract

Abstract:

We study the bound states to nonlinear Schrödinger equations with electromagnetic fields ih (∂ψ/∂t)=(h/i▽-A(x))²ψ + V(x)ψ-K(x)|ψ|^p-1ψ=0, on R₊×R^N. Let G (x)=[V (x)]^{(p+1)/(p-1)-N/2}[K(x)]^-2/(p-1) and suppose that G(x) has k local minimum points. For h > 0 small, we find multi-bump bound states ψ_h(x, t)=e^-iEt/h u_h(x) with u_h concentrating at the local minimum points of G(x) simultaneously as h → 0. The potentials V (x) and K(x) are allowed to be either compactly supported or unbounded at infinity.

Key words: Multi-bump solutions, nonlinear Schrödinger equation, electromagnetic fields, potentials compactly supported or unbounded at infinity

Na BA, Jinjun DAI. BOUND STATES OF SCHRÖDINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND VANISHING POTENTIALS[J].Acta mathematica scientia,Series B, 2017, 37(2): 405-424.

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