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 {-△（k）u:=-△u-（（（N-2）2）/4）U/︱X︱2-1/4 sum from i=1 to（k-1） u/（︱x︱2（In（i）R/︱x︱2））=f（x,u）,x ∈Ω,u=0,x ∈Ω,where 0 ∈ΩBa（0）RN,N≥3,ln（i）=i éj=1 ln（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 andnegative solutions are obtained by a variant mountain pass theorem.