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

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