A priori estimates for elliptic equations with reaction terms involving the function and its gradient

We study local and global properties of solutions of –$\mathrm{\Delta }$u = u p + M ||u| q in a domain $\mathrm{\Omega }$ of R N , in the range min{p, q} > 1 and M $\in$ R. We prove a priori estimates and existence or non-existence of ground states.