© Electronic Publishing House LARGE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS WITH NONLINEAR GRADIENT TERMS

Abstract. 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