LARGE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS WITH NONLINEAR GRADIENT TERMS

ALAN V. LAIR ； AIHUA W. WOOD

《International Journal of Mathematics and Mathematical Sciences》 1999卷 (1999/12) Pp. 869-883

https://doi.org/10.1155/S0161171299228694

摘要

We show that large positive solutions exist for the equation (P±) : Δu±|∇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 J_0^∞ tφ(t)dt ＜∞, whereφ(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.

關鍵字

Large solution ； elliptic equation ； existence of solution ； semilinear elliptic equation