Existence and concentration of semiclassical ground state solutions for the generalized Chern–Simons–Schrödinger system in [formula omitted

Abstract

This paper is concerned with the following singularly perturbed problem in H1(R2)−ε2Δu+V(x)u+A0(u(x))u+∑j=12Aj2(u(x))u=f(u),ε(∂1A2(u(x))−∂2A1(u(x)))=−12u2,∂1A1(u(x))+∂2A2(u(x))=0,εΔA0(u)=∂1(A2|u|2)−∂2(A1|u|2),where ε is a small parameter, V∈C(R2,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 ε0>0 determined by V and f such that for ε∈(0,ε0], the above problem admits a semiclassical ground state solution vˆε with exponential decay at infinity. We also establish a new concentration behaviour of {vˆε} as ε→0. In particular, our results are available to the nonlinearity f(u)∼|u|s−2u for s∈(4,6], which extend the existing results concerning the case f(u)∼|u|s−2u for s>6.