Ground state solutions for planar Schrödinger-Poisson system involving subcritical and critical exponential growth with convolution nonlinearity

Abstract

We study a class of planar Schrödinger-Poisson systems{−Δu+u+ϕu=(Iα⁎∫0uf(t)dt)f(u),x∈R2,Δϕ=u2,x∈R2, where Iα is the Riesz potential of order α∈(0,2). Under some suitable assumptions on f(u) with subcritical exponential growth, we consider the existence of positive Nehari-Pohoz̆aev type ground state solutions by using some new analytic techniques. Also, we obtain a positive ground state solution of Nehari type with the help of the non-Nehari manifold method when f(u) has critical exponential growth. Our results generalize and improve the ones in Du and Weth (2017) [20], Alves and Figueiredo (2019) [2], and some other related literatures as α→0+.