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
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.
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.