Ground states of nonlinear Schrödinger systems

Abstract

Abstract. This paper concerns the existence of positive radial ground states of the time-independent Schrödinger system ⎧ ⎨where n = 1, 2, 3, λ j > 0 and µ j > 0 for j = 1, 2, and β > 0. A result from Sirakov, Comm. Math. Phys. 271 (2007), 199-221, is improved.Consider the time-independent Schrödinger system (1)where n = 1, 2, 3, λ j > 0 and µ j > 0 for j = 1, 2, and β > 0. Solutions (u 1 (x), u 2 (x)) of (1) correspond to standing wave solutions (e iλ 1 t u 1 (x), e iλ 2 t u 2 (x)) of the timedependent system of 2 coupled nonlinear Schrödinger equationsThe system (2) stems from many physical problems, especially in nonlinear optics and in the Hartree-Fock theory for Bose-Einstein condensates; see, for example, [1,5,8,9,10,11,12,20,23]. Recently, (1) has attracted tremendous attention and has been studied extensively from the point of view of physics (see [1,10,11] for instance) as well as mathematics (see [2,3,4,13,14,15,16,18,19,21,22,24,25]). Solutions of (1) correspond to critical points of the energy functional (1)

Infinitely many solutions to a class of quasilinear Schrödinger system in RN

“…This class of system as (1.1), also known as Gross-Pitaevskii equations, have applications in many physical problems such as in nonlinear optics and in multispecies Bose-Einstein condensates, see [2]. Mathematical work for existence, non-existence, multiplicity, and qualitative properties of bound and ground state solutions has been done extensively in recent years in [3][4][5][6][7] and the references therein.…”

Existence of multiple solutions to a class of nonlinear Schrödinger system with external sources terms

“…It also models the Hartee-Fock theory for a double condensate, i.e., a binary mixture of Bose-Einstein condensates in two different hyperfine states |1 and |2 ( [15]). We refer the reader to [2,3,6,7,9,10,11,12,13,17,18,19,20,23], and the references therein for interesting existence of solutions or properties of solutions. The parameters µ i and β are the intraspecies and interspecies scattering lengths respectively.…”

“…Naturally, people concern nontrivial solutions(solutions with both components nonzero) of the system. In recent years, many interesting works have been devoted to searching ground states and bound states for this system, see [2,3,6,7,17,18,23] etc. and the references therein.…”

Positive ground states of coupled nonlinear Schrödingerequations on R<sup><italic>n</italic></sup>

“…的非平凡解、多解、径向解、非径向解和基态解的存在性, 以及解的集中性. 例如文献 [1][2][3][4][5][6][7][8][9][10][11]. 后来, Pomponio 和 Secchi [12] 利用 Pohozaev 等式证明了带有一般非线性项的系统   …”