Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy–Sobolev exponent and singular nonlinearity

摘要

In this paper, we study a class of critical elliptic problems of Kirchhoff type: [a+b(∫R3|∇u|2−μu2|x|2dx)2−α2](−Δu−μu|x|2)=|u|2∗(α)−2u|x|α+λf(x)|u|q−2u|x|β, where a, b  0, μ ∈ [0, 1/4), α, β ∈ [0, 2), and q ∈ (1, 2) are constants and 2^∗(α) = 6 − 2α is the Hardy–Sobolev exponent in ℝ³. For a suitable function f(x), we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard b  0 as a parameter to obtain the convergence property of solutions for the given problem as b ↘ 0⁺ by the mountain pass theorem and Ekeland’s variational principle.