Normalized solutions for the Chern-Simons-Schrödinger equation in R2

Abstract

In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear Chern-Simons-Schrödinger equations in, where h(s) = 1 / 2 ∫ 0s ru2(r) dr. To get such solutions we look for critical points of the energy functional on the constraints. When p = 4, we prove a sufficient condition for the nonexistence of constrain critical points of I on Sr(c) for certain c and get infinitely many minimizers of I on Sr(8π). For the value p ∈ (4,+∞) considered, the functional I is unbounded from below on Sr(c). By using the constrained minimization method on a suitable submanifold of Sr(c), we prove that for certain c > 0, I has a critical point on Sr(c). After that, we get an H1-bifurcation result of our problem. Moreover, by using a minimax procedure, we prove that there are infinitely many critical points of I restricted on Sr(c) for any.