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Abstract

We investigate the existence and multiplicity of positive solutions to the following 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 this paper 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.