Existence of Multiple Solutions for a Singular Elliptic Problem with Critical Sobolev Exponent

and Applied Analysis 3 The following Hardy-Sobolev inequality is due to Caffarelli et al. 12 , which is called Caffarelli-Kohn-Nirenberg inequality. There exist constants S1, S2 > 0 such that (∫ RN |x|−bp |u|pdx )p/p∗ ≤ S1 ∫ RN |x|−ap|∇u|pdx, ∀u ∈ C∞ 0 ( R N ) , 1.8 ∫ RN |x|− a 1 |u|dx ≤ S2 ∫ RN |x|−ap|∇u|pdx, ∀u ∈ C∞ 0 ( R N ) , 1.9 where p∗ Np/ N − pd is called the Sobolev critical exponent. In the present paper, we make the following assumptions: A1 f x ∈ L1 R, g1 ⋂ Lloc R N \ {0} for 1 < r < p, where g1 |x| a 1 rσ1 , σ1 p/ p − r ; A2 f x ∈ L2 R, g2 ⋂ Lloc R N\{0} for p < r < p∗, where g2 |x|brσ2 , σ2 p∗/ p∗−r . A3 h x ∈ L R, g3 ⋂ Lloc R N \{0} for p < s < p∗, where g3 |x|μbp , μ p∗/ p∗ −s . Then, we give some basic definitions. Definition 1.1. u ∈ X is said to be a weak solution of 1.1 if for any φ ∈ C∞ 0 R there holds ∫ RN ( |x|−ap|∇u|p−2∇u · ∇φ |u| p−2uφ |x| a 1 p ) dx ∫