Multiplicity and Stability of the Pohozaev Obstruction for Hardy-Schrödinger Equations with Boundary Singularity

Abstract

Let Ω \Omega be a smooth bounded domain in R n \mathbb {R}^n ( n ≥ 3 n\geq 3 ) such that 0 ∈ ∂ Ω 0\in \partial \Omega . We consider issues of non-existence, existence, and multiplicity of variational solutions in H 1 , 0 2 ( Ω ) H_{1,0}^2(\Omega ) for the borderline Dirichlet problem, { − Δ u − γ u | x | 2 − h ( x ) u a m p ; = a m p ; | u | 2 ⋆ ( s ) − 2 u | x | s     a m p ; in  Ω , u a m p ; = a m p ; 0 a m p ; on  ∂ Ω ∖ { 0 } , \begin{equation*} \left \{ \begin {array}{llll} -\Delta u-\gamma \frac {u}{|x|^2}- h(x) u &=& \frac {|u|^{2^\star (s)-2}u}{|x|^s} \ \ &\text {in } \Omega ,\\ \hfill u&=&0 &\text {on }\partial \Omega \setminus \{ 0 \} , \end{array} \right . \end{equation*} where 0 > s > 2 0>s>2 , 2 ⋆ ( s ) ≔ 2 ( n − s ) n − 2

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具有边界奇异性的Hardy-Schrödinger方程的Pohozaev阻塞的多重性和稳定性

让Ω\欧米茄be a smooth bounded域名in R n \ mathbb {R) ^ n ( n≥3 \ geq 3)这样的那个 0∈∂Ω0 \中\部分欧米茄。我们认为non-existence、存在的问题和multiplicity variational的解决方案在 H 1 , 0 2 ( Ω ) H_{1.0) ^ 2(\ω)》有点像Dirichlet问题,{ − Δ u − γ u | x | 2 − h ( x ) u a m p ;= m m p;| u | 2 ⋆ ( s ) − 2 u | x| s     a m p ;在   Ω , u a m p ;= m m p;零a m p;在   ∂ Ω ∖ { 0 } , \ 开始{equation *的左派\{\开始{}{llll阵}-三角洲u u -伽马\ frac {} {x | | - h (x) ^ 2的u & = & \ frac {| | u ^{2 ^ \星(s) - x的u} {| | \ & ^ s的短信{进来的,我是俄梅戛\ \ \ hfill u& = &0 & \短信上{}部分\ \ setminus \{0},我是俄梅戛end{阵列的\ coming right。\ end {equation *的地方 0 > s > 2 0 > s > , 2 ⋆ ( s ) ≔ 2 ( n − s ) n − 2 < mml: annotation

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