Existence of Positive Radial Solutions of General Quasilinear Elliptic Systems

Abstract

Let $\Omega \subset {\mathbb{R}}^n(n\ge 2)$ be either an open ball $B_R$ centred at the origin or the whole space. We study the existence of positive, radial solutions of quasilinear elliptic systems of the form \begin{equation*} \left\{ \begin{aligned} Δ_{p} u&=f_1(|x|)g_1(v)|\nabla u|^α &&\quad\mbox{ in } Ω, \\ Δ_{p} v&=f_2(|x|)g_2(v)h(|\nabla u|) &&\quad\mbox{ in } Ω, \end{aligned} \right. \end{equation*} where $\alpha \ge 0$, ${\Delta }_p$ is the $p$-Laplace operator, $p>1$, and for $i,j=1,2$ we assume $f_i,g_j,h$ are continuous, non-negative and non-decreasing functions. For functions $g_j$ which grow polynomially, we prove sharp conditions for the existence of positive radial solutions which blow up at $\partial B_R$, and for the existence of global solutions.