A planar Schrödinger–Newton system with Trudinger–Moser critical growth

Abstract

In this paper, we focus on the existence of positive solutions to the following planar Schrödinger–Newton system with general critical exponential growth -Δu+u+ϕu=f(u)inR2,Δϕ=u2inR2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} -\\Delta {u}+u+\\phi u =f(u)&{} \ ext{ in }\\,\\,\\mathbb {R}^2, \\\\ \\Delta {\\phi }=u^2 &{} \ ext{ in }\\,\\, \\mathbb {R}^2, \\end{array} \\right. \\end{aligned}$$\\end{document}where fin C^1(mathbb {R},mathbb {R}). We apply a variational approach developed in [36] to study the above problem in the Sobolev space H^1(mathbb {R}^2). The analysis developed in this paper also allows to investigate the relation between a Riesz-type of Schrödinger–Newton systems and a logarithmic-type of Schrödinger–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 frac{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.