Existence and concentration of semiclassical ground state solutions for the generalized Chern-Simons-Schrodinger system in H1 (R2)

This paper is concerned with the following singularly perturbed problem in H-1 (R-2) -epsilon(2) Delta u + V(x)u + A(0)(u(x))u + Sigma(2)(j=1) A(j)(2)(u(x))u = f(u), epsilon(partial derivative(1)A(2) (u(x)) - partial derivative(2)A(1) (u(x))) = -1/2u(2), partial derivative(1)A(1) (u(x)) + partial derivative(2)A(2) (u(x)) =0, epsilon Delta A(0)(u) = partial derivative(1)(A(2)vertical bar u vertical bar(2)) - partial derivative(2)(A(1)vertical bar u vertical bar(2)), where epsilon is a small parameter, V is an element of C(R-2, R) and f is an element of C(R, R). By using some new variational and analytic techniques joined with the manifold of Pohozaev-Nehari type, we prove that there exists a constant epsilon(0) > 0 determined by V and f such that for epsilon is an element of (0,epsilon(0)], the above problem admits a semiclassical ground state solution (v) over cap (epsilon) with exponential decay at infinity. We also establish a new concentration behaviour of {(v) over cap (epsilon)} as epsilon -> 0. In particular, our results are available to the nonlinearity f(u) similar to vertical bar u vertical bar(s-2)u for s is an element of (4,6], which extend the existing results concerning the case f(u) similar to vertical bar u vertical bar(s-2)u for s > 6. (C) 2019 Elsevier Ltd. All rights reserved.