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- Title
Schrödinger systems with quadratic interactions.
- Authors
Tian, Rushun; Wang, Zhi-Qiang; Zhao, Leiga
- Abstract
In this paper, we consider the existence and multiplicity of nontrivial solutions to a quadratically coupled Schrödinger system − Δ u + λ 1 u = μ 1 | u | u + β u v , in ℝ N , − Δ v + λ 2 v = μ 2 | v | v + β 2 u 2 , in ℝ N , where λ i , μ i and β are constants and 2 ≤ N ≤ 5 , i = 1 , 2. Such type of systems stem from applications in nonlinear optics, Bose–Einstein condensates and plasma physics. The existence (and nonexistence), multiplicity and asymptotic behavior of vector solutions of the system are established via variational methods. In particular, for multiplicity results we develop new techniques for treating variational problems with only partial symmetry for which the classical minimax machinery does not apply directly. For the above system, the variational formulation is only of even symmetry with respect to the first component u but not with respect to v , and we prove that the number of vector solutions tends to infinity as β tends to infinity.
- Subjects
BOSE-Einstein condensation; PLASMA physics; NONLINEAR optics; INFINITY (Mathematics); MULTIPLICITY (Mathematics); SCHRODINGER operator; HAMILTONIAN systems
- Publication
Communications in Contemporary Mathematics, 2019, Vol 21, Issue 8, pN.PAG
- ISSN
0219-1997
- Publication type
Article
- DOI
10.1142/S0219199718500773