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} \Delta_p u &= c_1|x|^{m_1} \cdot g_1(v) \cdot |\nabla u|^{\alpha} &\quad\mbox{ in } \mathbb R^n,\\ \Delta_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, $p>1$, $n\geq 2$, $c_1,c_2>0$ and $m_1, m_2, \alpha \geq 0$. 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.