Abstract: The author deals with the conditions of the existence of the singular elliptic equation -Δpu=｜u｜r-2u+( ｜u｜p(s)-2u）/（｜x｜s）+ f(x,u). An analysis on the (P.S.) sequence of the variational functionals corresponding to the equations is given and a local compactness result is obtained.By choosing special Mountain Pass theorem and energy estimate,the author obtains that if  the variational functionals corresponding to the equations satisfy local (P.S.) condition,there exsits a critical point,then the author proves the  exitence of positive solutions.

Key words: singularity, variational method, sobolev embedding, hardy inequality