Multiple positive solutions for a critical quasilinear equation via Morse theory
Multiple positive solutions for a critical quasilinear equation via Morse theory
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DOI：
10.1016/j.anihpc.2007.09.003
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发表时间：
2009-02
影响因子：
1.9
通讯作者：
S. Cingolani;Giuseppina Vannella
中科院分区：
文献类型：
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作者：
S. Cingolani;Giuseppina Vannella
− pu= λuq− 1+ up − 1 in Ω，u> 0 in Ω，u= 0 on Ω，其中Ω是RN中具有光滑边界的有界区域，N p2，1< p q< p，p = Np/（N-p），λ> 0是参数。利用Banach空间中的莫尔斯技巧，证明了存在λ λ ∈ 0，使得对任意λ∈（0，λ ∈），（Pλ）至少有P1（Ω）个解，其中Pt（Ω）是Ω的Poincaré多项式.对p ∈ 2，我们证明了对每个λ∈（0，λ <$），存在一列逼近（Pλ）的拟线性问题，每个问题至少有P1（Ω）个不同的正解.
− pu= λuq− 1+ up∗− 1 in Ω, u> 0 in Ω, u= 0 on∂ Ω, where Ω is a bounded domain in RN with smooth boundary, N⩾ p2, 1< p⩽ q< p∗, p∗= Np/(N− p), λ> 0 is a parameter. Using Morse techniques in a Banach setting, we prove that there exists λ∗> 0 such that, for any λ∈(0, λ∗),(Pλ) has at least P1 (Ω) solutions, possibly counted with their multiplicities, where Pt (Ω) is the Poincaré polynomial of Ω. Moreover for p⩾ 2 we prove that, for each λ∈(0, λ∗), there exists a sequence of quasilinear problems, approximating (Pλ), each of them having at least P1 (Ω) distinct positive solutions.