Existence and multiplicity of $L^2$-Normalized solutions for the periodic Schrödinger system of Hamiltonian type

TL;DR

The paper studies $L^2$-normalized solutions for the nonlinear Schrödinger system of Hamiltonian type with a periodic potential, under a mass constraint $\int_{\mathbb{R}^N}|z|^2dx=a^2$. It develops a strongly indefinite variational framework using a block operator $T$ with spectral decomposition into $E^+$ and $E^-$, and reduces the constrained problem to a controllable finite-dimensional setting via a map $\Phi$ and a perturbed functional to locate the Lagrange multiplier $\omega$. The authors prove the existence of a normalized solution for small $a$ and show a bifurcation point $\omega_a$ that tends to the spectral edge as $a\to0^+$; they also establish multiplicity for the scalar non-autonomous equation through a Lyusternik–Schnirelmann approach, obtaining infinitely many normalized solutions. The results extend the theory of normalized solutions to strongly indefinite, Hamiltonian-type systems with periodic potentials and provide a robust variational framework combining Lyapunov-Schmidt reduction, perturbation techniques, and LS category arguments.

Abstract

In this paper, we study the following nonlinear Schrödinger system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -Δu+V(x)u=\partial_v H(x,u,v)+ωv, \ x \in \mathbb{R}^N, \\ -Δv+V(x)v=\partial_u H(x,u,v)+ωu,\ x \in \mathbb{R}^N, \\ \displaystyle\int_{\mathbb{R}^N}|z|^2dx=a^2, \end{array}\right. \end{equation*} where the potential function $V(x)$ is periodic, $z:=(u,v):\mathbb{R}^N\rightarrow \mathbb{R}\times\mathbb{R}$, $ω\in \mathbb{R}$ appears as a Lagrange multiplier, $a>0$ is a prescribed constant. The existence and multiplicity of $L^2$-normalized solutions for the above Schrödinger system are obtained, and the combination of the Lyapunov-Schmidt reduction, a perturbation argument and the multiplicity theorem of Ljusternik-Schnirelmann is involved in the proof. In addition, a bifurcation result is also given.

Theorems & Definitions (17)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Proposition 3.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5