Monotonic convergence of positive radial solutions for general quasilinear elliptic systems

TL;DR

This paper investigates the asymptotic behavior of positive radial solutions to a class of quasilinear elliptic systems, proving their monotonic convergence to explicit power-law solutions under polynomial growth conditions.

Contribution

It establishes the monotonic convergence of solutions to explicit asymptotic power-law profiles for a broad class of quasilinear elliptic systems with polynomial growth functions.

Findings

Solutions asymptotically approach power-law functions.

Convergence of solution ratios is monotonic.

Results extend to more general 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.