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This article is devoted to the existence and multiplicity to the following singular elliptic equation with singular nonlinearities, Hardy-Sobolev critical exponent and weights: ⎪⎨ ⎪⎩ −Δu−μ u |x|2 = |u|p−2u |x|s +λ u |x|α |u| −β , x ∈ Ω, u > 0 x ∈ Ω, u = 0 x ∈ ∂Ω. where Ω is a smooth bounded domain in RN (N 3) , 0 ∈ Ω , λ > 0 , 0 μ < μN := (N−2) /4 , p = 2∗ (s) = 2(N− s)/(N−2) with 0 < s < 2 is the critical Hardy-Sobolev critical exponent, 0 α < N (p−1+β)/p , 0 < β < 1 and 2 < p 2∗ := 2N/(N−2) is the critical Sobolev exponent. By using the Nehari manifold and mountain pass theorem, the existence of at least four distinct solutions is obtained. Mathematics subject classification (2010): 35J66, 35J55, 35B40.

On singular elliptic equation with singular nonlinearities, Hardy-Sobolev critical exponent and weights