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 equation* \ aligned -Δu + (-Δ) ˢ u - μu{|x|² &= λ|u|^p-2 u + |u|^2^*-2 u in Ω RN, u &= 0 in RN Ω, aligned. equation* where Ω RN is a bounded domain with smooth boundary, 0 0, and μ (0, μ) where μ= (N-22) 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.
Malhotra et al. (Mon,) studied this question.