#### Document Type

Article

#### Publication Date

1999

#### 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 ∫ 0 ∞ t ϕ ( t ) d t < ∞ , 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 .

#### DOI

10.1155/S0161171299228694

#### Source Publication

International Journal of Mathematics and Mathematical Sciences

#### Recommended Citation

Alan V. Lair, Aihua W. Wood, "Large solutions of semilinear elliptic equations with nonlinear gradient terms", International Journal of Mathematics and Mathematical Sciences, vol. 22, Article ID 459151, 15 pages, 1999. https://doi.org/10.1155/S0161171299228694

## Comments

Published by Hindawi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (

CC BY 3.0)