Planar Schrödinger-Poisson system with zero mass potential and critical exponential growth

Abstract

This paper is concerned with the following planar Schrödinger-Poisson system with zero mass potential{−Δu+ϕu=Q(|x|)f(u),x∈R2,Δϕ=u2,x∈R2, where Q∈C((0,∞),(0,∞)) may be singular or unbound and f∈C(R,R) is of critical exponential growth and there is no monotonicity restriction on f(u)/u3. Based on the known Trudinger-Moser inequality in H0,rad1(B1), we establish a new version of Trudinger-Moser inequality within the working space associated with the energy functional relevant to the aforementioned problem, which is different from Albuquerque et al. (2021) [5]. By combining the variational methods and delicate estimates, we prove the existence of a non-trivial mountain-pass solution to the above system under mild assumptions on Q and f.