Abstract. The equation considered in this paper is $$ \left (a(t)\phi _p(x')\right )' + b(t)\phi _p(x') + c(t)\phi _p(x) = 0. $$ Here, $\phi _p(x') = |x'|^{p-2}x'$ with $p > 1$. The coefficients $a(t)$, $b(t)$ and $c(t)$ are not assumed to be positive. The main purpose is to present sharp conditions for the global asymptotic stability of the zero solution of a system equivalent to this differential equation. Sufficient conditions are also given for the zero solution to be globally attractive. Our results are new even in the linear case ($p = 2$). Some suitable examples are included to illustrate the main theorem. Global phase portraits are also attached for a deeper understanding. Finally, certain changes of variable are used to broaden the application of our results.

AMS Subject Classification (1991): 34D05, 34D23; 37B25, 37B55

Received November 28, 2006, and in revised form August 25, 2007. (Registered under 6453/2009.)