The effect of domain topology on the number of positive solutions for singular elliptic problems involving the Caffarelli–Kohn–Nirenberg inequalities

In this paper we consider the critical singular equation involving the Caffarelli–Kohn–Nirenberg inequalities of the type (P1) ⎧⎪ ⎪⎨⎪ ⎪⎩ −div(|x|−2a∇u) −μ u |x|2(1+a) = λ|u|q−2u |x|dD + |u|p−2u |x|bp , x∈ Ω, u>0, x∈ Ω, u = 0, x∈ ∂Ω. Here Ω is a bounded domain with smooth boundary in RN and contains 0 in its interior, 0 μ < (√μ¯ − a)2, μ¯ = (N−2 2 )2, N 3, a b < a + 1, a d < a + 1, p = p(a, b) 2N N−2(1+a−b) , D = D(a, d) 2N N−2(1+a−d) , λ is a positive parameter and 2 q