Large Solutions of Semilinear Elliptic Equations with Nonlinear Gradient Terms

We show that large positive solutions exist for the equation (P±) :∆u±|∇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|=t p(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.