We study the Hamiltonian elliptic system 0.1 \(\begin{aligned} \left\{ \begin{aligned} -\Delta u&= \lambda |v|^{r-1}v +|v|^{p-1}v \qquad&\hbox {in} \ \ \Omega ,\\ -\Delta v&= \mu |u|^{s-1}u +|u|^{q-1}u \qquad&\hbox {in} \ \ \Omega ,\\ u&>0, \ v>0 \qquad \,&\hbox {in} \ \ \Omega ,\\ u&=v = 0 \qquad \quad&\hbox {on} \quad \partial \Omega , \end{aligned} \right. \end{aligned}\) where \(\Omega \subset \mathbb {R}^N\) is a smooth bounded domain, \(\lambda \) and \( \mu \) are nonnegative parameters and \(r,s,p,q>0\) . Our study includes the case in which the nonlinearities in (0.1) are concave near the origin and convex near infinity, and we focus on the region of non-negative pairs of parameters \((\lambda ,\mu )\) that guarantee existence and multiplicity of solutions of (0.1). In particular, we show the existence of a strictly decreasing curve \(\lambda _*(\mu )\) on an interval \([0, \mu ]\) with \(\lambda _*(0)> 0, \lambda _*(\mu ) = 0\) and such that the system has two solutions for \((\lambda ,\mu )\) below the curve, one solution for \((\lambda , \mu )\) on the curve and no solution for \((\lambda , \mu )\) above the curve. A similar statement holds reversing \(\lambda \) and \(\mu \) .
