Existence Result for Non-linearly Perturbed Hardy-Schr\"odinger Problems: Local and Non-local cases

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a smooth bounded domain having zero in its interior $0\in \mathrm{\Omega }\mathrm{.}$ We fix $0<\alpha \le 2$ and $0\le s<\alpha \mathrm{.}$ We investigate a sufficient condition for the existence of a positive solution for the following perturbed problem associated with the Hardy-Schr"odinger operator $ L_{\gamma,\alpha,}: = ({-}{ \Delta})^{\frac{\alpha}{2}}- \frac{\gamma}{|x|^{\alpha}}$on$\Omega:$ \begin{equation*} \left{\begin{array}{rl} \displaystyle ({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}} - \lambda u= {\frac{u^{2_{\alpha}^(s)-1}}{|x|^s}}+ h(x) u^{q-1} & $\text{in }$ {\Omega}\ u=0 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, & $\text{in }$ $\mathbb{R}$^n \setminus \Omega, \end{array}\right. \end{equation} where $2_{\alpha }^{\ast }(s):=\frac{2(n-s)}{n-\alpha },$ $\lambda \in \mathbb{R}$, $h\in C^0(\bar{\mathrm{\Omega }}),$ $h\ge 0,$ $q\in (2,2_{\alpha }^{\ast })$ with $2_{\alpha }^{\ast }:=2_{\alpha }^{\ast }(0),$ and $\gamma <{\gamma }_H(\alpha ),$ the latter being the best constant in the Hardy inequality on ${\mathbb{R}}^n\mathrm{.}$ We prove that there exists a threshold ${\gamma }_{crit}(\alpha )$ in $( - \infty, \gamma_H(\alpha)) $ such that the existence of solutions of the above problem is guaranteed by the non-linear perturbation $(i\mathrm{.}e\mathrm{.},h(x)u^{q-1})$ whenever $\gamma \le {\gamma }_{crit}(\alpha ),$ while for $\gamma_{crit}(\alpha)<\gamma <\gamma_H(\alpha)$, it is determined by a subtle combination of the geometry of the domain and the size of the nonlinearity of the perturbations.