Normalized solutions for a Hamiltonian elliptic system on RN

This paper is concerned with the existence of normalized solutions for the following class of Hamiltonian elliptic systems: {align*}\{{array}{ll}- Δ u = λ u + |v|q-2v in RN, \\ - Δ v = λ v + |u|ᵖ⁻²u in RN, \\ ∫R^N(|u|^2 + |v|^2) = m,{array}.{align*} where m > 0 and 2 < p,q < 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 H¹(RN) L²(RN) , which persists also if we restrict ourselves to a radial setting. Our main result is novel for this class of systems.