Multiple and sign-changing solutions for nonlinear elliptic equation with critical potential and critical parameter

Some embedding inequalities in Hardy-Sobolev space are proved. Furthermore, by the improved inequalities and the linking theorem, in a new k-order Sobolev-Hardy space, we obtain the existence of sign-changing solutions for the nonlinear elliptic equation {-Δ(k)u:=-Δu-(N-2)24u|x|2-14∑i=1k-1u|x|2(In(i)R/|x|)2=f(x,u)x∈Ω,u=0,x∈∂Ω, where 0∈Ω⊂Ba(0)⊂ℝN,N≥3, In(i)=∏j=1iIn(j) and R=ae(k-1), where e(0) = 1, e(j) = ee(j−1) for j≥ 1, ln(1)= ln, ln(j) = ln ln(j−1) for j ≥ 2. Besides, positive and negative solutions are obtained by a variant mountain pass theorem.