International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 4, Pages 869-883

Large solutions of semilinear elliptic equations with nonlinear gradient terms

Alan V. Lair and Aihua W. Wood

Department of Mathematics and Statistics, Air Force Institute of Technology/ENC, 2950 P Street, Wright-Patterson AFB 45433-7765, OH, USA

Received 19 June 1998

Copyright © 1999 Alan V. Lair and Aihua W. Wood. 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 show that large positive solutions exist for the equation (P±):Δu±|u|q=p(x)uγ in ΩRN(N3) for appropriate choices of γ>1,q>0 in which the domain Ω is either bounded or equal to RN. The nonnegative function p is continuous and may vanish on large parts of Ω. If Ω=RN, then p must satisfy a decay condition as |x|. For (P+), the decay condition is simply 0tϕ(t)dt<, where ϕ(t)=max|x|=tp(x). For (P), we require that t2+βϕ(t) be bounded above for some positive β. Furthermore, we show that the given conditions on γ and p are nearly optimal for equation (P+) in that no large solutions exist if either γ1 or the function p has compact support in Ω.