Existence of Positive Radial Solutions of General Quasilinear Elliptic Systems

TL;DR

This paper establishes sharp conditions for the existence of positive radial solutions to a class of quasilinear elliptic systems involving the p-Laplace operator, considering solutions that blow up at the boundary or exist globally.

Contribution

It provides new sharp criteria for the existence of positive radial solutions to complex quasilinear elliptic systems with polynomial growth functions.

Findings

Conditions for solutions blowing up at boundary are characterized.

Global solutions exist under specific growth conditions.

Results apply to systems involving the p-Laplace operator with non-decreasing functions.

Abstract

Let $\Omega\subset\mathbb R^{n}\ (n\geq2)$ 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} \Delta_{p} u&=f_1(|x|)g_1(v)|\nabla u|^{\alpha} &&\quad\mbox{ in } \Omega, \\ \Delta_{p} v&=f_2(|x|)g_2(v)h(|\nabla u|) &&\quad\mbox{ in } \Omega, \end{aligned} \right. \end{equation*} where $\alpha\geq 0$, $\Delta_{p}$ 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.