Large solutions of semilinear elliptic equations with nonlinear gradient terms

Alan V. Lair and Aihua W. Wood

International Journal of Mathematics and Mathematical Sciences, 1999, vol. 22, 1-15

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 .

Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:459151

DOI: 10.1155/S0161171299228694

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