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We consider the Hardy–Schrödinger operatorL  γ   :=    -   Δ     n-   γ  ⁢   V  2{L_{\gamma}:=-\Delta_{\mathbb{B}^{n}}-\gamma{V_{2}}}    on the Poincaré ball model of the hyperbolic space  n{\mathbb{B}^{n}}    (n  ≥  3{n\geq 3}   ). HereV  2{V_{2}}    is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e.,V  2   ⁢   (  r  )    ∼   1   r  2{V_{2}(r)\sim\frac{1}{r^{2}}}   . As in the Euclidean setting,L  γ{L_{\gamma}}    is positive definite wheneverγ  &lt;(   n  -  2   )   2   4{\gamma&lt;\frac{(n-2)^{2}}{4}}   , in which case we exhibit explicit solutions for the critical equationL  γ   ⁢  u   =    V    2  *   ⁢   (  s  )⁢   u2  *   ⁢   (  s  )    -  1{L_{\gamma}u=V_{2^{*}(s)}u^{2^{*}(s)-1}}    in  n   ,{\mathbb{B}^{n},}    where0  ≤  s  &lt;  2{0\leq s&lt;2}   ,2  *   ⁢   (  s  )    =    2  ⁢   (   n  -  s   )n  -  2{2^{*}(s)=\frac{2(n-s)}{n-2}}   , andV    2  *   ⁢   (  s  ){V_{2^{*}(s)}}    is a weight that behaves like1   r  s{\frac{1}{r^{s}}}    around 0. In dimensionsn  ≥  5{n\geq 5}   , the equationL  γ   ⁢  u   -   λ  ⁢  u    =    V    2  *   ⁢   (  s  )⁢   u2  *   ⁢   (  s  )    -  1{L_{\gamma}u-\lambda u=V_{2^{*}(s)}u^{2^{*}(s)-1}}    in a domain Ω of  n{\mathbb{B}^{n}}    away from the boundary but containing 0 has a ground state solution, whenever0  &lt;  γ  ≤    n  ⁢   (   n  -  4   )    4{0&lt;\gamma\leq\frac{n(n-4)}{4}}   , andλ  &gt;n  -  2    n  -  4    ⁢   (n  ⁢   (   n  -  4   )    4   -  γ   ){\lambda&gt;\frac{n-2}{n-4}(\frac{n(n-4)}{4}-\gamma)}   . On the other hand, in dimensions 3 and 4, the existence of solutions depends on whether the domain has a positive “hyperbolic mass” a notion that we introduce and analyze therein.

Mass and Extremals Associated with the Hardy–Schrödinger Operator on Hyperbolic Space