Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation

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• 作者：Luiz Gustavo Farah ; ^{lgfarah@gmail.com} ; Carlos M. Guzmá ; n ^{carlos.guz.j@gmail.com}
• 关键词：35Q55 ; 35P25 ; 35B40
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：15 April 2017
• 年：2017
• 卷：262
• 期：8
• 页码：4175-4231
• 全文大小：642 K
• 卷排序：262

文摘

The purpose of this work is to study the 3D focusing inhomogeneous nonlinear Schrödinger equationiut+Δu+|x|−b|u|2u=0,iut+Δu+|x|−b|u|2u=0, where 0<b<1/20<b<1/2. Let Q   be the ground state solution of −Q+ΔQ+|x|−b|Q|2Q=0−Q+ΔQ+|x|−b|Q|2Q=0 and sc=(1+b)/2sc=(1+b)/2. We show that if the radial initial data u0u0 belongs to H1(R3)H1(R3) and satisfies E(u0)scM(u0)1−sc<E(Q)scM(Q)1−scE(u0)scM(u0)1−sc<E(Q)scM(Q)1−sc and ‖∇u0‖L2sc‖u0‖L21−sc<‖∇Q‖L2sc‖Q‖L21−sc, then the corresponding solution is global and scatters in H1(R3)H1(R3). Our proof is based in the ideas introduced by Kenig–Merle [1] in their study of the energy-critical NLS and Holmer–Roudenko [2] for the radial 3D cubic NLS.