Normalized solutions for a Hamiltonian elliptic system on $\mathbb{R}^N$

This paper is concerned with the existence of normalized solutions for the following class of Hamiltonian elliptic systems: \begin{align*}\left\{\begin{array}{ll}- \Delta u = \lambda u + |v|^{q-2}v \quad \text{in } \mathbb{R}^{N}, \ - \Delta v = \lambda v + |u|^{p-2}u \quad \text{in } \mathbb{R}^{N}, \ \displaystyle\int_{\mathbb{R}^N}(|u|^2 + |v|^2) = m,\end{array}\right.\end{align*} where $m \gt 0$ and $2 \lt p,q \lt 2^{*}=2N/(N-2)$ . We prove that a normalized solution exists for different ranges of $p,q$ . A typical feature of this class of problems is that the associated energy functionals are strongly indefinite; that is, the domain has a saddle-point geometry in which both positive and negative subspaces of the quadratic form are infinite-dimensional. Another difficulty is the lack of the compact embedding $H^{1}(\mathbb{R}^N) \hookrightarrow L^{2}(\mathbb{R}^N)$ , which persists also if we restrict ourselves to a radial setting. Our main result is novel for this class of systems.