Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

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摘要 In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy-Sobolev-Maz＇ya term：-Δu-λ u/｜y｜2 = （｜u｜pt-1u）/｜y｜t ＋ μf（x）, x∈ Ω,where Ω is a bounded domain in RN（N ≥ 2）, 0 ∈ Ω, x = （y, z） ∈ Rk ×RN-k and pt = （N＋2-2t）/（N-2）（0 ≤ t ≤ 2）. For f（x） ∈ C1（Ω）／{0}, we show that there exists a constant μ＊〉0 such that the problem possesses at least two positive solutions if μ ∈ （0, μ＊） and at least one positive solution if μ = μ＊. Furthermore, there are no positive solutions if μ ∈ （μ＊,＋∞）. In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy-Sobolev-Maz＇ya term：-Δu-λ u/｜y｜2 = （｜u｜pt-1u）/｜y｜t ＋ μf（x）, x∈ Ω,where Ω is a bounded domain in RN（N ≥ 2）, 0 ∈ Ω, x = （y, z） ∈ Rk ×RN-k and pt = （N＋2-2t）/（N-2）（0 ≤ t ≤ 2）. For f（x） ∈ C1（Ω）／{0}, we show that there exists a constant μ＊〉0 such that the problem possesses at least two positive solutions if μ ∈ （0, μ＊） and at least one positive solution if μ = μ＊. Furthermore, there are no positive solutions if μ ∈ （μ＊,＋∞）.

作者 Shuang Jie PENG Jing YANG

机构地区 School of Mathematics and Statistics

出处《Acta Mathematica Sinica,English Series》 SCIE CSCD 2015年第6期893-912,共20页 数学学报（英文版）

基金 Supported by NSFC(Grant No.11301204) the Ph D specialized grant of the Ministry of Education of China(Grant No.20110144110001) the excellent doctorial dissertation cultivation grant from Central China Normal University(Grant No.2013YBZD15)

关键词 Hardy-Sobolev-Maz＇ya inequality Mountain Pass Lemma positive solutions subsolutionand supersolution Hardy-Sobolev-Maz＇ya inequality, Mountain Pass Lemma, positive solutions, subsolutionand supersolution

分类号 O175.25 [理学—基础数学]

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Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

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