Large solution of a semilinear elliptic problem

We show that large positive solutions exist for the following equationΔu+|∇u|q=p(x)f(u)(p+)in Ω ⊆ RN (N ≥ 3) 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∫0∞rϕ(r)dr<∞,whereϕ(r)=max⁡|x|=rp(x)as|x|→∞.Furthermore, we show that the given conditions on p are nearly optimal for equation (p+).