On the Chern-Simons-Schrödinger Equation with Critical Exponential Growth

This paper is concerned with the following Chern-Simons-Schrödinger equation −Δu+V(|x|)u+(∫|x|∞h(s)su2(s)ds+h2(|x|)|x|2)u=a(|x|)f(u)inR2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- \Delta u + V(\left| x \right|)u + \left({\int_{\left| x \right|}^\infty {{{h(s)} \over s}{u^2}(s)ds + {{{h^2}(\left| x \right|)} \over {{{\left| x \right|}^2}}}}} \right)u = a(\left| x \right|)f(u)\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^2},$$\end{document} where h(s)=∫0sl2u2(l)dl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(s) = \int_0^s {{l \over 2}{u^2}(l)dl} $$\end{document}, V, a: ℝ+ → ℝ are radially symmetric potentials and the nonlinearity f: ℝ → ℝ is of subcritical or critical exponential growth in the sense of Trudinger-Moser. We give some new sufficient conditions on f to obtain the existence of nontrivial solutions or ground state solutions. In particular, some new estimates and techniques are used to overcome the difficulty arising from the critical growth of Trudinger-Moser type.