Large Solutions of Semilinear Elliptic Equations with Nonlinear Gradient Terms
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 .
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
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