Abstract In this paper, we investigate the following Chern-Simons-Schrodinger system { − Δ u + V ( x ) u + A 0 u + A 1 2 u + A 2 2 u = f ( x , u ) , x ∈ R 2 , ∂ 1 A 2 − ∂ 2 A 1 = − 1 2 u 2 , ∂ 1 A 1 + ∂ 2 A 2 = 0 , ∂ 1 A 0 = A 2 u 2 , ∂ 2 A 0 = − A 1 u 2 . where V is the potential, ∂ 1 = ∂ ∂ x 1 , ∂ 2 = ∂ ∂ x 2 for x = ( x 1 , x 2 ) ∈ R 2 , A j : R 2 → R is the gauge field ( j = 0 , 1 , 2 ) and the nonlinearity f ( x , s ) ∈ C ( R 2 × R , R ) behaves like e 4 π s 2 as | s | → + ∞ . If V and f are both asymptotically periodic at infinity, we prove the existence of positive ground state solutions by combining the Nehari manifold methods with the Trudinger-Moser inequality.

中文翻译：

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具有临界增长的渐近周期陈-西蒙-薛定谔系统的正基态解

摘要 在本文中，我们研究了以下陈-西蒙斯-薛定谔系统 { − Δ u + V ( x ) u + A 0 u + A 1 2 u + A 2 2 u = f ( x , u ) , x ∈ R 2 , ∂ 1 A 2 − ∂ 2 A 1 = − 1 2 u 2 , ∂ 1 A 1 + ∂ 2 A 2 = 0 , ∂ 1 A 0 = A 2 u 2 , ∂ 2 A 0 = − A 1 u 2 . 其中 V 是势能，∂ 1 = ∂ ∂ x 1 ，∂ 2 = ∂ ∂ x 2 对于 x = ( x 1 , x 2 ) ∈ R 2 ，A j : R 2 → R 是规范场 ( j = 0 , 1 , 2 ) 和非线性 f ( x , s ) ∈ C ( R 2 × R , R ) 表现为 e 4 π s 2 为 | | → + ∞ 。如果 V 和 f 在无穷远处都是渐近周期的，我们通过将 Nehari 流形方法与 Trudinger-Moser 不等式相结合来证明正基态解的存在。