On the structure of positive radial solutions to an equation containing a p-Laplacian with weight

Let A,B:(0,∞)↦(0,∞) be two given weight functions and consider the equation(P)-divA(|x|)|∇u|p-2∇u=B(|x|)|u|q-2u,x∈Rn,where q>p>1. By considering positive radial solutions to this equation that are bounded, we are led to study the initial value problem-a(r)|u′|p-2u′′=b(r)(u+)q-1,r∈(0,∞),u(0)=α>0,limr→0a(r)|u′(r)|p-1=0,where a(r)=r(N-1)A(r) and b(r)=r(N-1)B(r). By means of two key functions m and Bq defined below, we obtain several new results that allow us to classify solutions to this initial value problem as being respectively crossing, slowly decaying, or rapidly decaying. We also generalize several results in Clément et al. (Asymptotic Anal. 17 (1998) 13-29), Kawano et al. (Funkcial. Ekvac 36 (1993) 121-145), Yanagida and Yotsutani (Arch. Rational Mech. Anal. 124 (1993) 239-259), Yanagida and Yotsutani (J. Differential Equations 115 (1995) 477-502), Yanagida and Yotsutani (Arch. Rational Mech. Anal. 134 (1996) 199-226).