Coercive elliptic systems with gradient terms

In this paper we give a classification of positive radial solutions of the following system:$\Delta u=v^{m},\quad\Delta v=h(|x|)g(u)f(|\nabla u|),$in the open ball ${B_{R}}$, with ${m>0}$, and f, g, h nonnegative nondecreasing continuous functions. In particular, we deal with both explosive and bounded solutions. Our results involve, as in [27], a generalization of the well-known Keller–Osserman condition, namely, ${\int_{1}^{\infty}(\int_{0}^{s}F(t)\,dt)^{-m/(2m+1)}\,ds<\infty}$, where ${F(t)=\int_{0}^{t}f(s)\,ds}$. Moreover, in the second part of the paper, the p-Laplacian version, given by ${\Delta_{p}u=v^{m}}$, ${\Delta_{p}v=f(|\nabla u|)}$, is treated. When ${p\geq 2}$, we prove a necessary condition for the existence of a solution with at least a blow up component at the boundary, precisely ${\int_{1}^{\infty}(\int_{0}^{s}F(t)\,dt)^{-m/(mp+p-1)}s^{(p-2)(p-1)/(mp+p-1)}% \,ds<\infty}$.