Zhou Xin LI; Yao Tian SHEN
We consider the p-Laplacian-like equation with critical exponent:-div (a(|Du|~p)|Du|~(p-2)Du)=|u|~(p~(*-2))u+λf(x,u),u∈W_0~(1,p)(Ω),whereΩ∈R~N (N≥2) is a bounded smooth domain and a is a smooth function from R~+ to R.The solutions are obtained by variational methods,the analysis of Palais-Smale sequences requires suitable generalizations of the techniques involved in the study of the corresponding quasilinear problem with lack of compactness.Using the concentration compactness principle of Lions,the result that the associated functional I_λsatisfies the (PS)_c con- dition is proved.Applying the Clark's critical theory and the properties of genus,the existence of infinitely many solutions of the problem is obtained.Furthermore,the existence of a special eigenfunction whenλ>0 small enough is proved.