We investigate the existence and multiplicity of positive solutions to the following critical exponent problem driven by the superposition of the Laplacian and the fractional Laplacian with Hardy potential
\begin{equation*}
\left\{
\begin{aligned}
-\Delta u + (-\Delta)^s u - \mu \frac{u}{|x|^2} &= \lambda |u|^{p-2} u + |u|^{2^*-2} u \quad \text{in } \Omega \subset \mathbb{R}^N, \\
u &= 0 \quad \text{in } \mathbb{R}^N \setminus \Omega,
\end{aligned}
\right.
\end{equation*}
where \( \Omega \subset \mathbb{R}^N \) is a bounded domain with smooth boundary, $ 0 < s < 1 $, $ 1 < p < 2^* $, with $ 2^* = \frac{2N}{N-2} $, $ \lambda > 0 $, and $ \mu \in (0, \bar{\mu}) $ where $\bar \mu = \left( \frac{N-2}{2} \right)^2$.

The aim of the talk is twofold. First, we establish uniform asymptotic estimates for solutions of the problem by means of a suitable transformation. Then, according to the value of the exponent \(p\), we analyze three distinct cases and prove the existence of a positive solution. Moreover, in the sublinear regime \(1 < p < 2\), we demonstrate the existence of multiple positive solutions for small perturbations of the fractional Laplacian.