The Least-Energy Sign-Changing Solutions for Planar Schrödinger-Newton System with an Exponential Critical Growth

Abstract

In 1990, the notion of critical growth in ℝ2 was introduced by Adimurthi and Yadava. Soon afterwards, de Figueiredo, Miyagaki and Ruf also studied the solvability of elliptic equations in dimension two. They treated the problems in the subcritical and the critical case. In 2019, Alves and Figueieredo proved the existence of a positive solution for a planar Schrodinger-Poisson system, where the nonlinearity is a continuous function with exponent critical growth. Also, in 2020, Chen and Tang investigated the planar Schrodinger-Poisson system with the critical growth nonlinearity. Under the axially symmetric assumptions, they obtained infinitely many pairs solutions and ground states. In this work, motivated by the works mentioned above and Z. Angew. Math. Phys., 66 (2015) 3267-3282, Z. Angew. Math. Phys., 67 (2016) 102, 18, we study the planar Schrodinger-Newton system with a Coulomb potential where the nonlinearity f is autonomous nonlinearity which belongs to C1 and satisfies super-linear at zero and exponential critical at infinity. Moreover, we need that f satisfies the Nehari type monotonic condition. We obtain a least-energy sign-changing solution via the variational method. To be more precise, we define the sign-changing Nehari manifold. And the least-energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.