On some nonlinear elliptic PDEs with Sobolev–Hardy critical exponents and a Li–Lin open problem
On some nonlinear elliptic PDEs with Sobolev–Hardy critical exponents and a Li–Lin open problem
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DOI：
10.1007/s00526-015-0844-z
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发表时间：
2015-02
影响因子：
2.1
通讯作者：
G. Cerami;X. Zhong;W. Zou
中科院分区：
文献类型：
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作者：
G. Cerami;X. Zhong;W. Zou
设Ω Ω为R^ N，\, N≥3，RN， N≥3中的C^ 1 c1开有界定域，其中0∈Ω. 0∈Ω¯。我们考虑以下涉及Hardy-Sobolev临界指数的问题：(P)\qquad\left {(P) Δ u+ λ up| x| s 1+ u 2 * (s2)-1| x| s 2= 0在Ω中，u (x)> 0在Ω中；u (x)在∂Ω上= 0，其中0≤s_1< 2,0≤s_2< 2,2 ^*(s_2):= 2 (N-s_2) n - 2,0≠λ∈R, 1≤p≤2^*(s_1)-1 0≤s 1< 2,0≤s 2< 2,2∗(s 2):= 2 (N-s 2) n - 2,0≠λ∈R, 1≤p≤2∗(s 1)-1，并选择指数和参数对应于(p)之前未研究过的情况。我们证明了正解的存在性，在某些情况下，正解也被证明是基态。我们注意到，我们给出了Li和Lin （Arch Ration Mech, 203(3): 943-968, 2012）提出的问题的第一个部分答案。
Let Ω Ω be a C^ 1 C 1 open bounded domain in R^ N,\, N ≥ 3, RN, N≥ 3, with 0 ∈ ̄ Ω. 0∈ Ω¯. We consider the following problem involving Hardy–Sobolev critical exponents:(P)\qquad\left {(P) Δ u+ λ up| x| s 1+ u 2∗(s 2)-1| x| s 2= 0 in Ω, u (x)> 0 in Ω; u (x)= 0 on∂ Ω, where 0 ≤ s_1< 2, 0 ≤ s_2< 2, 2^*(s_2):= 2 (N-s_2) N-2, 0 ≠ λ ∈ R, 1 ≤ p ≤ 2^*(s_1)-1 0≤ s 1< 2, 0≤ s 2< 2, 2∗(s 2):= 2 (N-s 2) N-2, 0≠ λ∈ R, 1≤ p≤ 2∗(s 1)-1, and with choices of exponents and parameters corresponding to cases in which (P) has not been before investigated. We prove the existence of positive solutions, which, in some cases, are also shown to be ground states. We remark that we give a first partial answer to a question proposed by Li and Lin (Arch Ration Mech Anal 203 (3): 943–968, 2012).