Positive Solutions and Infinitely Many Solutions for a Weakly Coupled System
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
POSITIVE SOLUTIONS AND INFINITELY MANY SOLUTIONS FOR A WEAKLY COUPLED SYSTEM∗
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Xueliang DUAN (
)†
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China E-mail : [email protected]
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Gongming WEI (
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China E-mail : [email protected]
°7)
Haitao YANG (
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China E-mail : [email protected] Abstract We study a Schr¨ odinger system with the sum of linear and nonlinear couplings. Applying index theory, we obtain infinitely many solutions for the system with periodic potentials. Moreover, by using the concentration compactness method, we prove the existence and nonexistence of ground state solutions for the system with close-to-periodic potentials. Key words
coupled Schr¨ odinger system, ground state solution, infinitely many solutions, concentration compactness principle
2010 MR Subject Classification
1
35A15; 35J60
Introduction and Main Results
Recently, significant effort has been focused on coupled Schr¨ odinger systems (see [4, 5, 8, 9, 11, 13, 19, 20, 22]). These systems appear in various branches of mathematical physics, e.g., Kerr-like photorefractive media in optics, the propagation in birefringent optical fibers and Bose-Einstein condensates (see [6, 7, 18]). A solution (u, v) is called a nonzero ground state solution if ω = (u, v) 6= (0, 0) is a critical point of J such that J (u, v) = inf N J , where J is the corresponding functional and N is the corresponding Nehari manifold. A ground state solution (u, v) is nontrivial if ω = (u, v), u 6= 0 and v = 6 0. It is well known that there are linear and nonlinear forms of couplings for coupled Schr¨odinger systems. Ambrosetti et al. [1] studied the following linearly coupled Schr¨odinger system: −∆u + u = (1 + a (x))|u|p−1 u + λv in RN , 1 (1.1) −∆v + v = (1 + a2 (x))|v|p−1 v + λu in RN , ∗ Received
November 21, 2018; revised December 2, 2019. author: Xueliang DUAN.
† Corresponding
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
where lim ai (x) = 0, i = 1, 2. The existence of positive ground state solutions of (1.1) has |x|→∞
been proven by concentration compactness methods. Maia et. al., in [11], studied a system with nonlinear couplings. For the case of typical nonlinearities, the existence of a nontrivial ground state solution was proven by the Pohozaev manifold method. We consider a Schr¨ odinger system with the sum of linear and nonlinear couplings as follows: q−2 q N −∆u + V1 (x)u = f1 (u) + |u| u|v| + λ(x)v in R ,
−∆v + V2 (x)v = f2 (v) + |v|q−2 v|u|q + λ(x)u 1
in RN ,
(1.2)
N
u, v ∈ H (R ),
∗ where N > 2, 2∗ = N2N −2 , 2 < 2q < p < 2 . Furthermore, fi , i = 1, 2 satisfies the following conditions: (F1 ) for fi ∈ C 1 (R) and fi (u) 6≡ 0, there exists a constant c > 0 such that
|fi′ (u)| ≤ c(1 + |u|p−2 ); (F2 ) fi (u) = −fi (−u) and
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