Existence and concentration behavior of solutions for the logarithmic Schrodinger-Poisson system with steep potential

In this paper, we study the following logarithmic Schrodinger-Poisson system: { -delta u + lambda V (x)u + q(x)phi u = u log u(2), in R-3,-delta phi = q(x)u(2), in R-3, where lambda > 0, V (x) E C(R-3,R) and q(x) > 0 for all x E R-3. Under the suitable conditions on potential V (x) and q(x), by proceeding a new penalization scheme to the nonlocal term phi, combining with a new version of Minimax method, we prove the existence of positive solution u(lambda) E H-1(R-3) of the above system for lambda > 0 large enough. Moreover, we also investigate the concentration behavior of lu(lambda)} as lambda -> +infinity.