A positive ground state solution of asymptotically periodic Chern-Simons-Schrödinger systems with critical growth

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In this paper, we investigate the following Chern-Simons-Schrödinger system{−Δu+V(x)u+A0u+A12u+A22u=f(x,u),x∈R2,∂1A2−∂2A1=−12u2,∂1A1+∂2A2=0,∂1A0=A2u2,∂2A0=−A1u2, where V is the potential, ∂1=∂∂x1,∂2=∂∂x2 for x=(x1,x2)∈R2, Aj:R2→R is the gauge field (j=0,1,2) and the nonlinearity f(x,s)∈C(R2×R,R) behaves like e4πs2 as |s|→+∞. If V and f are both asymptotically periodic at infinity, we prove the existence of positive ground state solutions by combining the Nehari manifold methods with the Trudinger-Moser inequality.