Journal of Applied Mathematics and Stochastic Analysis 
Volume 10 (1997), Issue 1, Pages 95-108
doi:10.1155/S1048953397000105

Positive and oscillatory radial solutions of semilinear elliptic equations

Shaohua Chen,1 William R. Derrick,1 and Joseph A. Cima2

 1University of Montana, Department of Mathematics, Missoula 59812, MT, USA
2University of North Carolina, Department of Mathematics, Chapel Hill 27599, NC, USA
Received 1 September 1995; Revised 1 February 1996

Abstract

We prove that the nonlinear partial differential equation Δu+f(u)+g(|x|,u)=0, in  ℝn,n≥3, with u(0)>0, where f and g are continuous, f(u)>0 and g(|x|,u)>0 for u>0, and limu→0+f(u)uq=B>0, for 1<q<n/(n−2), has no positive or eventually positive radial solutions. For g(|x|,u)≡0, when n/(n−2)≤q<(n+2)/(n−2) the same conclusion holds provided 2F(u)≥(1−2/n)uf(u), where F(u)=∫0uf(s)ds. We also discuss the behavior of the radial solutions for f(u)=u3+u5 and f(u)=u4+u5 in ℝ3 when g(|x|,u)≡0.