Existence and concentration behavior of solutions for the logarithmic Schrodinger-Poisson system via penalization method

In this paper, we study the following logarithmic Schrodinger-Poisson system{-epsilon 2 triangle u + V (x)u- phi u = u log u(2), in R-3,-epsilon 2 triangle phi = u(2), in R-3, where epsilon is a small positive parameter and V(x) E C(R-3,R). Under the local condition on potential V(x), we prove the existence of positive solution u epsilon E H1(R-3) of above system for epsilon > 0 small enough by combining the variational method with a penalization scheme. Moreover, we also investigate the concentration behavior of fu epsilon} as epsilon-} 0.