Ground State Solution for the Logarithmic Schrödinger–Poisson System with Critical Growth

In this paper we investigate a class of logarithmic Schrödinger–Poisson system with critical growth which are from physically relevant backgrounds. Under suitable assumptions on the potential V(x), we obtain the existence of ground state solution for \(\varepsilon >0\) $\epsilon >0$ sufficiently small via using a constrained minimization on a Nehari manifold and some new techniques. Specially, by restricting the least energy c to a suitable interval, we overcome some difficulties brought by the critical term \(u^5\) $u^5$ for restoring the compactness of the minimizing sequence of the least energy c.