Large solutions of semilinear elliptic equations with nonlinear gradient terms

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.