Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz’ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ*> 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).