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

Abstract

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 : = - Δ u - ( N - 2 ) 2 4 u | x | 2 - 1 4 ∑ i = 1 k - 1 u | x | 2 ( In ( i ) R / | x | ) 2 = f ( x , u ) x ∈ Ω , u = 0 , x ∈ ∂ Ω , where 0 ∈ Ω ⊂ B a ( 0 ) ⊂ ℝ N , N ≥ 3 , In ( i ) = ∏ j = 1 i In ( j ) and R = a e ( k - 1 ) , where e (0) = 1, e ( j) = e e ( 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.