The role of the green's function in a non-linear elliptic equation involving the critical Sobolev exponent
The role of the green's function in a non-linear elliptic equation involving the critical Sobolev exponent
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DOI：
10.1016/0022-1236(90)90002-3
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发表时间：
1990-03
影响因子：
1.7
通讯作者：
O. Rey
中科院分区：
文献类型：
--
作者：
O. Rey
本文考虑（P ε）型非线性椭圆问题：− Δu= u（N+ 2）（N− 2）+ εu，u> 0 on Ω; u= 0 on Ω，其中Ω是RN中的光滑有界区域，N≥ 4，ε> 0.证明了当ε→ 0时，如果（P ε）的解u ε集中在一点上，则该点不可能在Ω的边界上，而是绿色函数正则部分的临界点.反之，我们证明了对N≥ 5和绿色函数正则部分的任意非退化临界点x 0，当ε→ 0时，存在（P ε）的解集中在x 0附近.
This paper is concerned with non-linear elliptic problems of the type (P ε):− Δu= u (N+ 2)(N− 2)+ εu, u> 0 on Ω; u= 0 on∂ Ω, where Ω is a smooth and bounded domain in R N, N≥ 4, and ε> 0. We show that if the u ε are solutions of (P ε) which concentrate around a point as ε→ 0, then this point cannot be on the boundary of Ω and is a critical point of the regular part of the Green's function. Conversely, we show that for N≥ 5 and any non-degenerate critical point x 0 of the regular part of the Green's function, there exist solutions of (P ε) concentrating around x 0 as ε→ 0.