Normalized solutions to the Chern-Simons-Schrödinger system under the nonlinear combined effect

We investigate normalized solutions to a class of Chern-Simons-Schrödinger systems with combined nonlinearities f(u) = ∣u∣p−2u + μ∣u∣q−2u in ℝ2,where μ ∈ {± 1} and 2 < p,q < ∞. The solutions correspond to critical points of the underlying energy functional subject to the L2-norm constraint, namely, ∫ℝ2|u|2dx=c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int {_{{^2}}} {\left| u \right|^2}dx = c$$\end{document} for c > 0 given. Of particular interest is the competing and double L2-supercritical case, i.e., μ = −1 and min{p, q} > 4. We prove several existence and multiplicity results depending on the size of the exponents p and q. It is worth emphasizing that some of them are also new even in the study of the Schrödinger equations. In addition, the asymptotic behaviors of the solutions and the associated Lagrange multipliers λ as c → 0 are described.