Existence and multiplicity of normalized solutions for the nonlinear Chern–Simons–Schrödinger equations

Abstract

In this paper, we prove the existence and multiplicity results of solutions with prescribed L 2-norm for a class of nonlinear Chern-Simons-Schrödinger equations in R 2 2 −ˆR 2 F (u) constrained on the L 2-spheres S r (c) = u ∈ H 1 r (R 2) : u 2 2 = c. Here, c > 0 and F (s) :=´s 0 f (t) dt. Under some mild assumptions on f , we show that critical points of E κ unbounded from below on S r (c) exist for certain c > 0. In addition, we establish the existence of infinitely many critical points {u κ n } of E κ on S r (c) provided that f is odd. Finally, we regard κ as a parameter and and present a convergence property of u κ n as κ ց 0. These results improve and generalize the existing ones in the literature.