A planar Schrodinger-Newton system with Trudinger-Moser critical growth

Abstract

In this paper, we focus on the existence of positive solutions to the following planar Schrodinger-Newton system with general critical exponential growth $-\Delta u + u + \phi u = f (u) in R^2, \Delta \phi = u^2 in R^2 $, where $f$ is an element of $ C^1( R, R)$. We apply a variational approach developed in [36] to study the above problem in the Sobolev space $H^1(R^2)$. The analysis developed in this paper also allows to investigate the relation between a Riesz-type of Schrodinger-Newton systems and a logarithmic-type of Schrodinger-Poisson systems. Furthermore, this approach can overcome some difficulties resulting from either the nonlocal term with sign-changing and unbounded logarithmic integral kernel, or the critical nonlinearity, or the lack of monotonicity of $ f(t)/t(3)$. We emphasize that it seems much difficult to use the variational framework developed in the existed literature to study the above problem.In this paper, we focus on the existence of positive solutions to the following planar Schrodinger-Newton system with general critical exponential growth $-\Delta u + u + \phi u = f (u) in R^2, \Delta \phi = u^2 in R^2 $, where $f$ is an element of $ C^1( R, R)$. We apply a variational approach developed in [36] to study the above problem in the Sobolev space $H^1(R^2)$. The analysis developed in this paper also allows to investigate the relation between a Riesz-type of Schrodinger-Newton systems and a logarithmic-type of Schrodinger-Poisson systems. Furthermore, this approach can overcome some difficulties resulting from either the nonlocal term with sign-changing and unbounded logarithmic integral kernel, or the critical nonlinearity, or the lack of monotonicity of $ f(t)/t(3)$. We emphasize that it seems much difficult to use the variational framework developed in the existed literature to study the above problem.