In this paper, we study the following nonlinear Schr\”{o}dinger system of
Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -\Delta
u+V(x)u=\partial_v H(x,u,v)+\omega v, \ x \in \mathbb{R}^N, \\ -\Delta
v+V(x)v=\partial_u H(x,u,v)+\omega u,\ 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,
$z:=(u,v):\mathbb{R}^N\rightarrow \mathbb{R}\times\mathbb{R}$, $\omega\in
\mathbb{R}$ arises as a Lagrange multiplier, $a>0$ is a prescribed constant.
The main result in this paper establishes the existence and multiplicity of
$L^2$-normalized solutions for the above nonlinear Schr\”{o}dinger system with
a class of non-autonomous nonlinearity $H(x,u,v)$. The proofs combine
Lyapunov-Schmidt reduction, perturbation argument and the multiplicity theorem
of Ljusternik-Schnirelmann. Inoltre, we obtain bifurcation results of this
problem.

Questo articolo esplora i giri e le loro implicazioni.

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