含临界指数的类p-Laplacian方程无穷多解的存在性

Existence of Infinitely Many Solutions for p-Laplacian-Like Equation with Critical Exponent

考虑如下一类含临界指数的类p-Laplacian方程-div(a(|Du|~p)|Du|^(p-2)Du)=:-- |u|^(p^*-2)u+λf(x,u),u∈W_0^(1,p)(Ω),其中Ω∈R^N(N≥2)为有界光滑区域,a:R^+→R为连续函数.由于问题失去紧性,对Palais-Smale序列的分析需要一点技巧.本文利用Lions的集中紧原理,证明了相应泛函I_λ满足(PS)_c条件,再应用Clark临界点定理和亏格的性质,证明了方程无穷多解的存在性.进一步,得到当λ充分小时一个特殊的特征函数的存在性.

We consider the p-Laplacian-like equation with critical exponent：-div（a（｜Du｜^p）｜Du｜^p-2Du）=｜u｜^p＊-2u＋λf（x,u）,u∈W0^1,p（Ω）,, where Ω∈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 Iλ satisfies 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 when λ 〉 0 small enough is proved.