Normalized solutions to the Chern-Simons-Schrödinger system: the supercritical case

Abstract

We are concerned with the existence of normalized solutions for a class of generalized Chern-Simons-Schrödinger type problems with supercritical exponential growth $$-\Delta u+\lambda u+A_{0}u+\sum _{j=1}^2{A}_j^{2}u=f(u),{\partial }_1{A}_2-{\partial }_2{A}_1=-\frac{1}{2}|u{|}^2,{\partial }_1{A}_1+{\partial }_2{A}_2=0,{\partial }_1{A}_0=A_2|u{|}^2,{\partial }_2{A}_0=-A_1|u{|}^2,{\int }_{{\mathbb{R}}^2}|u{|}^{2}dx=a^2,$$ where $a\ne 0$, $\lambda \in \mathbb{R}$ is known as the Lagrange multiplier and $f\in C^1(\mathbb{R})$ denotes the nonlinearity that fulfills the supercritical exponential growth in the Trudinger-Moser sense at infinity. Under suitable assumptions, combining the constrained minimization approach together with the homotopy stable family and elliptic regularity theory, we obtain that the problem has at least a ground state solution.

39 pages