Existence of axially symmetric solutions for the planar Schrödinger–Newton equations with critical exponential growth

Abstract

This paper is concerned with the following planar Schrödinger–Newton equation −Δu+V(x)u+12π(ln(|⋅|)∗|u|p)|u|p−2u=f(x,u),x∈R2,where p≥2, V∈C(R2,[0,∞)) is axially symmetric and f∈C(R2×R,R) is of critical exponential growth in the sense of Trudinger–Moser. Under mild assumptions, we obtain the existence of axially symmetric solutions to the above equation for p≥2 by using some new techniques. In particular, our theorem not only extends the result of Cao et al. (2021), where f(x,u) has polynomial growth on u, but covers the case p=2 in Chen and Tang (2020).