Existence and concentration of semiclassical ground state solutions for the generalized Chern–Simons–Schrödinger system in H1(R2). (August 2019) - Chen, Sitong; Zhang, Binlin; Tang, Xianhua
Abstract: This paper is concerned with the following singularly perturbed problem in H 1 ( R 2 ) − ε 2 Δ u + V ( x ) u + A 0 ( u ( x ) ) u + ∑ j = 1 2 A j 2 ( u ( x ) ) u = f ( u ), ε ( ∂ 1 A 2 ( u ( x ) ) − ∂ 2 A 1 ( u ( x ) ) ) = − 1 2 u 2, ∂ 1 A 1 ( u ( x ) ) + ∂ 2 A 2 ( u ( x ) ) = 0, ε Δ A 0 ( u ) = ∂ 1 ( A 2 | u | 2 ) − ∂ 2 ( A 1 | u | 2 ), where ε is a small parameter, V ∈ C ( R 2, R ) and f ∈ C ( R, R ) . By using some new variational and analytic techniques joined with the manifold of Pohoz̆aev–Nehari type, we prove that there exists a constant ε 0 > 0 determined by V and f such that for ε ∈ ( 0, ε 0 ], the above problem admits a semiclassical ground state solution v ˆ ε with exponential decay at infinity. We also establish a new concentration behaviour of { v ˆ ε } as ε → 0 . In particular, our results are available to the nonlinearity f ( u ) ∼ | u | s − 2 u for s ∈ ( 4, 6 ], which extend the existing results concerning the case f ( u ) ∼ | u | s − 2 u for s > 6 .