Monotonic convergence of positive radial solutions for general quasilinear elliptic systems

Abstract

We study the asymptotic behavior of positive radial solutions for quasilinear elliptic systems that have the form \begin{equation*} \left\{ \begin{aligned} Δ_p u &= c_1|x|^{m_1} \cdot g_1(v) \cdot |\nabla u|^α &\quad\mbox{ in } \mathbb R^n,\\ Δ_p v &= c_2|x|^{m_2} \cdot g_2(v) \cdot g_3(|\nabla u|) &\quad\mbox{ in } \mathbb R^n, \end{aligned} \right. \end{equation*} where ${\Delta }_p$ denotes the $p$-Laplace operator, $p>1$, $n\ge 2$, $c_1,c_2>0$ and $m_1,m_2,\alpha \ge 0$. For a general class of functions $g_j$ which grow polynomially, we show that every non-constant positive radial solution $(u,v)$ asymptotically approaches $(u_0,v_0)=(C_{\lambda }|x{|}^{\lambda },C_{\mu }|x{|}^{\mu })$ for some parameters $\lambda ,\mu ,C_{\lambda },C_{\mu }>0$. In fact, the convergence is monotonic in the sense that both $u/u_0$ and $v/v_0$ are decreasing. We also obtain similar results for more general systems.