Journal of Applied Mathematics and Stochastic Analysis
Volume 10 (1997), Issue 1, Pages 95-108

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

Copyright © 1997 Shaohua Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We prove that the nonlinear partial differential equation Δu+f(u)+g(|x|,u)=0, in  n,n3, with u(0)>0, where f and g are continuous, f(u)>0 and g(|x|,u)>0 for u>0, and limu0+f(u)uq=B>0, for 1<q<n/(n2), has no positive or eventually positive radial solutions. For g(|x|,u)0, when n/(n2)q<(n+2)/(n2) the same conclusion holds provided 2F(u)(12/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 3 when g(|x|,u)0.