Critical planar Schrödinger–Poisson equations: existence, multiplicity and concentration

In this paper, we are concerned with the study of the following 2-D Schrödinger–Poisson equation with critical exponential growth

$$\begin{aligned} -\varepsilon ^2\Delta u+V(x)u+\varepsilon ^{-\alpha }(I_\alpha *|u|^q)|u|^{q-2}u=f(u), \end{aligned}$$

where \(\varepsilon >0\) is a parameter, \(I_\alpha \) is the Riesz potential, \(0<\alpha <2\), \(V\in {\mathcal {C}}({{\mathbb {R}}}^2,{{\mathbb {R}}})\), and \(f\in {\mathcal {C}}({{\mathbb {R}}},{{\mathbb {R}}})\) satisfies the critical exponential growth. By variational methods, we first prove the existence of ground state solutions for the above system with the periodic potential. Then we obtain that there exists a positive ground state solution of the above system concentrating at a global minimum of V in the semi-classical limit under some suitable conditions. Meanwhile, the exponential decay of this ground state solution is detected. Finally, we establish the multiplicity of positive solutions by using the Ljusternik–Schnirelmann theory.