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This paper is concerned with the quasi-linear equation with critical SobolevHardy exponent where Ω RN(N ≥ 3) is a smooth bounded domain, 0 ∈Ω, 0 ≤ s < p, 1 < p < N,p* (s) :=p(N- s)/N-p is the critical Sobolev-Hardy exponent, λ> 0,p ≤ r < p* ,p* := Np/N-p