Positive solutions for the Hardy–Sobolev–Maz'ya equation with Neumann boundary condition

Let Ω   be a bounded domain with a smooth C2C2 boundary in RN=Rk×RN−kRN=Rk×RN−k (N≥3N≥3), 0∈∂Ω0∈∂Ω, and ν denotes the unit outward normal vector to boundary ∂Ω  . We are concerned with the Neumann boundary problem: −Δu−μu|y|2=|u|pt−1u|y|t+f(x,u), u>0u>0, x∈Ωx∈Ω, ∂u∂ν+α(x)u=0, x∈∂Ω∖{0}x∈∂Ω∖{0}. Using the Mountain Pass Lemma without (PS) condition and the strong maximum principle, we establish certain existence result of the positive solutions.