Positive Solution for Critical Elliptic Problem with Hardy Potential and Logarithmic Perturbation

In this article, we are interested in studying a positive solution of the following problem: { − Δ u − μ u | x | 2 = | u | 2 * − 2 u + α u + β u log u 2 in Ω , u = 0 on ∂ Ω , where Ω ⊂ R N is an open bounded set, α , β ∈ R , N ≥ 3 , μ ∈ R + be such that 0 < μ < ( N − 2 ) 2 / 4 , and 2 * = 2 N / ( N − 2 ) is the critical Sobolev exponent. The novelty of our work lies in investigation of crucial energy estimates involving logarithmic terms, which are essential to overcome the lack of compactness. With the help of these estimates, we obtain our results in two cases for α ∈ R