Existence and non-existence results for a class of systems under concave-convex nonlinearities

Abstract In this work, we are interested in studying the following class of problems: (𝒫 λμ ) \begin{align} \left\{ \begin{array}{ll} -\Delta u=f_\lambda(x,u,v)& \text{in}~~\Omega\\ -\Delta v=g_\mu(x,u,v) & \text{in}~~\Omega\\ 0\not\equiv u\geq 0,\,\,0\not\equiv v\geq 0& \text{in}~~\Omega\\ u=v=0&\text{on}~~\partial\Omega \end{array} \right. \end{align} where Ω is a bounded domain in $\mathbb{R}^N$ , λ > 0, µ > 0, $t\mapsto f_\lambda(x,t,t)$ and $t\mapsto g_\mu(x,t,t)$ have concave-convex type nonlinearities. We present results related to the existence and non-existence of solutions for problem $(\mathcal{P}_{\lambda\mu})$ .