涉及临界Hardy-Sobolev指数的椭圆方程与系统（非极限情形）
偏微分方程分析
2015-05-28 v1
摘要
设 Ω ⊂ R N \Omega\subset \R^N Ω ⊂ R N N ≥ 3 N\geq 3 N ≥ 3 0 ∈ ∂ Ω 0\in \partial\Omega 0 ∈ ∂ Ω ∂ Ω \partial\Omega ∂ Ω 0 0 0 C 2 C^2 C 2 H ( 0 ) H(0) H ( 0 ) { − Δ u + λ ∗ u ∣ x ∣ σ 0 − λ 1 ∣ u ∣ 2 ∗ ( s 1 ) − 2 u ∣ x ∣ s 1 = λ 1 ∣ x ∣ s 2 ∣ u ∣ α − 2 u ∣ v ∣ β in Ω , − Δ v + μ ∗ v ∣ x ∣ η 0 − μ 1 ∣ v ∣ 2 ∗ ( s 1 ) − 2 v ∣ x ∣ s 1 = μ 1 ∣ x ∣ s 2 ∣ u ∣ α ∣ v ∣ β − 2 v in Ω , ( u , v ) ∈ D 0 1 , 2 ( Ω ) × D 0 1 , 2 ( Ω ) , \begin{cases} -\Delta u+\lambda^*\frac{u}{|x|^{\sigma_0}}-\lambda_1 \frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=\lambda \frac{1}{|x|^{s_2}}|u|^{\alpha-2}u|v|^\beta\quad &\hbox{in}\;\Omega,\\ -\Delta v+\mu^*\frac{v}{|x|^{\eta_0}}-\mu_1 \frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=\mu \frac{1}{|x|^{s_2}}|u|^{\alpha}|v|^{\beta-2}v\quad &\hbox{in}\;\Omega,\\ (u,v)\in D_{0}^{1,2}(\Omega)\times D_{0}^{1,2}(\Omega), \end{cases} ⎩ ⎨ ⎧ − Δ u + λ ∗ ∣ x ∣ σ 0 u − λ 1 ∣ x ∣ s 1 ∣ u ∣ 2 ∗ ( s 1 ) − 2 u = λ ∣ x ∣ s 2 1 ∣ u ∣ α − 2 u ∣ v ∣ β − Δ v + μ ∗ ∣ x ∣ η 0 v − μ 1 ∣ x ∣ s 1 ∣ v ∣ 2 ∗ ( s 1 ) − 2 v = μ ∣ x ∣ s 2 1 ∣ u ∣ α ∣ v ∣ β − 2 v ( u , v ) ∈ D 0 1 , 2 ( Ω ) × D 0 1 , 2 ( Ω ) , in Ω , in Ω , 0 ≤ σ 0 , η 0 , s 2 < 2 , s 1 ∈ ( 0 , 2 ) ; 0\leq \sigma_0, \eta_0, s_2<2, s_1\in (0,2); 0 ≤ σ 0 , η 0 , s 2 < 2 , s 1 ∈ ( 0 , 2 ) ; λ ∗ ≠ 0 , μ ∗ ≠ 0 , λ 1 > 0 , μ 1 > 0 , λ μ > 0 \lambda^*\neq 0, \mu^*\neq 0, \lambda_1>0, \mu_1>0, \lambda\mu>0 λ ∗ = 0 , μ ∗ = 0 , λ 1 > 0 , μ 1 > 0 , λ μ > 0 α , β > 1 \alpha,\beta>1 α , β > 1 α + β ≤ 2 ∗ ( s 2 ) \alpha+\beta \leq 2^*(s_2) α + β ≤ 2 ∗ ( s 2 ) 2 ∗ ( s ) : = 2 ( N − s ) N − 2 2^*(s):=\frac{2(N-s)}{N-2} 2 ∗ ( s ) := N − 2 2 ( N − s )
引用
@article{arxiv.1505.07392,
title = {On Elliptic Equations and Systems involving critical Hardy-Sobolev exponents (non-limit case)},
author = {Zhong Xuexiu and Zou Wenming},
journal= {arXiv preprint arXiv:1505.07392},
year = {2015}
}
备注
64 pages