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

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 >...

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Bibliographic Details
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
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Summary: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.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-023-01244-7