International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 4, Pages 869-883
doi:10.1155/S0161171299228694
Large solutions of semilinear elliptic equations with nonlinear gradient terms
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.
Abstract
We show that large positive solutions exist for the equation (P±):Δu±|∇u|q=p(x)uγ in Ω⫅RN(N≥3) 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 ∫0∞tϕ(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 γ≤1 or the function p has compact support in Ω.