具有混合反应的非自治拟线性椭圆方程的正超解

IF 0.8 4区 数学 Q2 MATHEMATICS
Asadollah Aghajani, Vicenţiu D. Rădulescu
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引用次数: 0

摘要

给出了在Ω中求解椭圆型问题-Δ p u+b(x)|∇u| pq q+1 =c(x)u q的新liouvile型定理的一种简单方法,其中Ω是一个N≥p>1且q≥p-1的外域。在q≠p-1的情况下,我们主要处理b(x)=|x| a, c(x)=λ|x| σ的势,其中λ>0,且a,σ∈x。我们证明了在参数p,q,a,σ的某些范围内不存在正超解,这是最优的。当q=p-1时,考虑上述问题具有一般权值b(x)≥0,c(x)>0,并假设c(x)-b p (x) p p >0,对于较大的|x|,我们也允许lim |x|→∞[c(x)-b p (x) p p]=0。b和c的权值是无界的。我们证明了如果这个方程有一个正的超解,那么势必须满足一个不依赖于超解的相关微分不等式。建立了与τ =lim sup |x|→∞|x|b(x)≤∞有关的正超解不存在的充分条件。证明中的一个关键因素是与p-拉普拉斯算子相关的广义hardy型不等式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction
We provide a simple method for obtaining new Liouville-type theorems for positive supersolutions of the elliptic problem -Δ p u+b(x)|∇u| pq q+1 =c(x)u q in Ω, where Ω is an exterior domain in ℝ N with N≥p>1 and q≥p-1. In the case q≠p-1, we mainly deal with potentials of the type b(x)=|x| a , c(x)=λ|x| σ , where λ>0 and a,σ∈ℝ. We show that positive supersolutions do not exist in some ranges of the parameters p,q,a,σ, which turn out to be optimal. When q=p-1, we consider the above problem with general weights b(x)≥0, c(x)>0 and we assume that c(x)-b p (x) p p >0 for large |x|, but we also allow the case lim |x|→∞ [c(x)-b p (x) p p ]=0. The weights b and c are allowed to be unbounded. We prove that if this equation has a positive supersolution, then the potentials must satisfy a related differential inequality not depending on the supersolution. We also establish sufficient conditions for the nonexistence of positive supersolutions in relationship with the values of τ:=lim sup |x|→∞ |x|b(x)≤∞. A key ingredient in the proofs is a generalized Hardy-type inequality associated to the p-Laplace operator.
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来源期刊
CiteScore
1.70
自引率
0.00%
发文量
92
审稿时长
1 months
期刊介绍: The Annales de l’Institut Fourier aim at publishing original papers of a high level in all fields of mathematics, either in English or in French. The Editorial Board encourages submission of articles containing an original and important result, or presenting a new proof of a central result in a domain of mathematics. Also, the Annales de l’Institut Fourier being a general purpose journal, highly specialized articles can only be accepted if their exposition makes them accessible to a larger audience.
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