Existence and Concentration of Semi-classical Ground State Solutions for Chern–Simons–Schrödinger System

• Published in: Qualitative theory of dynamical systems Vol. 20; no. 2
• Main Authors: Wang, Lin-Jing, Li, Gui-Dong, Tang, Chun-Lei
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.07.2021; Springer Nature B.V
• Subjects: Difference and Functional Equations; Dynamical Systems and Ergodic Theory; Ground state; Mathematical analysis; Mathematics; Mathematics and Statistics; Variational methods; Concentration; Variational methods; Chern–Simons–Schrödinger system; Semi-classical solution; Ground state solutions
• ISSN: 1575-5460; 1662-3592
• DOI: 10.1007/s12346-021-00480-y

In this paper, we consider the equation - ε 2 Δ u + V ( x ) u + A 0 ( u ) + A 1 2 ( u ) + A 2 2 ( u ) u = f ( u ) in H 1 ( R 2 ) , where ε is a small parameter, V is the external potential, A i ( i = 0 , 1 , 2 ) is the gauge field and f ∈ C ( R , R ) is 5-superlinear growth. By using variational methods and analytic technique, we prove that this system possesses a ground state solution u ε . Moreover, our results show that, as ε → 0 , the global maximum point x ε of u ε must concentrate at the global minimum point x 0 of V .