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 Δpu=c1|x|m1⋅g1v⋅|∇u|αin Rn,Δpv=c2|x|m2⋅g2v⋅g3|∇u|in Rn, where Δp denotes the p-Laplace operator, p > 1, n⩾2 , c1,c2>0 and m1,m2,α⩾0 . For a general class of functions gj which grow polynomially, we show that every non-constant positive radial solution (u, v) asymptotically approaches (u0,v0)=(Cλ|x|λ,Cμ|x|μ) for some parameters λ,μ,Cλ,Cμ>0 . In fact, the convergence is monotonic in the sense that both u/u0 and v/v0 are decreasing. We also obtain similar results for more general systems.