Generalized Chern–Simons–Schrödinger System with Sign-Changing Steep Potential Well: Critical and Subcritical Exponential Case

• Published in: The Journal of geometric analysis 2023-06, Vol.33 (6), Article 185
• Main Authors: Pomponio, Alessio, Shen, Liejun, Zeng, Xiaoyu, Zhang, Yimin
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Eigenvalues; Existence theorems; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Infinity; Mathematics; Mathematics and Statistics
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-023-01244-7

We consider the following generalized Chern–Simons–Schrödinger system in R 2 - Δ u + ( λ V ( x ) - μ ) u + A 0 u + ∑ j = 1 2 A j 2 u = f ( x , u ) , ∂ 1 A 2 - ∂ 2 A 1 = - 1 2 | u | 2 , ∂ 1 A 1 + ∂ 2 A 2 = 0 , A 1 ∂ 1 u + A 2 ∂ 2 u = 0 , ∂ 1 A 0 = A 2 | u | 2 , ∂ 2 A 0 = - A 1 | u | 2 , where λ > 0 is a parameter, V ∈ C ( R 2 , R + ) has a potential well Ω ≜ int V - 1 ( 0 ) , μ ∉ { μ j } j = 1 ∞ with { μ j } j = 1 ∞ being the eigenvalues of ( - Δ , H 0 1 ( Ω ) ) . If μ < μ 1 , we obtain the existence and concentrating behaviour of ground state solutions for λ sufficiently large under some suitable assumptions on f having critical exponential growth at infinity. Let μ ∈ ( μ j 0 , μ j 0 + 1 ) for some j 0 ∈ N + , employing the Morse theory, linking argument and the symmetric mountain-pass theorem, we are concerned with the existence and multiplicity of nontrivial solutions for sufficiently large λ when f has a subcritical exponential growth at infinity.