Existence and concentration of ground state solutions for critical Schrodinger-Poisson system with steep potential well

In this paper, we consider the following Schrodinger-Poisson system { -Delta u + (1 + mu g(x))u + phi u = vertical bar u vertical bar(4)u + lambda vertical bar u vertical bar(q-2)u, in R-3, -Delta phi = u(2, )in R-,(3) where q is an element of (3, 6) and lambda, mu>0 are positive parameters. Since f (u): =vertical bar u vertical bar(4)u + lambda vertical bar u vertical bar(q-2) u with q is an element of (3, 4] does not satisfy the (AR) condition. Thus, we construct Nehari-Pohozaev-Palais-Smale sequence to overcome the boundedness of sequence. As q is an element of (4, 6), the boundedness of sequence is easily obtained. We need (g(1)) and (g(2)) to prove that c(mu) < 1/3 S-3/2 independent of mu. Furthermore, we utilize the definition of the set of solutions to seek a ground state solution. Besides, the concentration behavior of the ground state solution is also described as mu -> infinity. (C) 2020 Elsevier Inc. All rights reserved.