Large solutions of semilinear elliptic equations with nonlinear gradient terms

Abstract

We show that large positive solutions exist for the equation ( P ±) : Δ u ± |∇ u | q = p ( x ) u γ in Ω ⫅ R N ( N ≥ 3) for appropriate choices of γ > 1, q > 0 in which the domain Ω is either bounded or equal to R N . The nonnegative function p is continuous and may vanish on large parts of Ω . If Ω = R N , then p must satisfy a decay condition as | x | → ∞ . For ( P +), the decay condition is simply , where ϕ ( t ) = max | x |= t p ( x ). For ( P −), we require that t 2+ β ϕ ( 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 .