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

. This article is devoted to the existence and multiplicity to the following singular ellip- tic equation with singular nonlinearities, Hardy-Sobolev critical exponent and weights: where Ω is a smooth bounded domain in R N ( N (cid:2) 3 ) , 0 ∈ Ω , λ > 0, 0 (cid:3) μ < μ N : = ( N − 2 ) 2 / 4, p = 2 ∗ ( s ) = 2 ( N − s ) / ( N − 2 ) with 0 < s < 2 is the critical Hardy-Sobolev critical exponent, 0 (cid:3) α < N ( p − 1 + β ) / p , 0 < β < 1 and 2 < p (cid:3) 2 ∗ : = 2 N / ( 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.