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

In this paper, we study the following logarithmic Schrödinger-Poisson system{−ε2Δu+V(x)u−ϕu=ulog⁡u2,inR3,−ε2Δϕ=u2,inR3, where ε is a small positive parameter and V(x)∈C(R3,R). Under the local condition on potential V(x), we prove the existence of positive solution uε∈H1(R3) of above system for ε>0 small enough by combining the variational method with a penalization scheme. Moreover, we also investigate the concentration behavior of {uε} as ε→0.