一类Sobolev空间紧嵌入定理

Imbedding theorem for a kind of Sobolev space

利用Hlder插值不等式论证了仅需Sobolev空间有界弱收敛子序列在某个L^(p)(R^(N))空间上强收敛。借助更弱位势函数自身性质、有界区域上经典的Sobolev紧嵌入定理,巧妙地将全空间划分为3个特殊区间,证明了带有更弱位势函数的一类Sobolev空间紧嵌入定理。有效地解决了带有位势函数的椭圆偏微分方程解的存在性因工作空间失去紧性所产生的困难。

The Hlder interpolating inequality was used to prove that the Sobolev compact imbedding theorem holds if and only if the bounded sequence has some strong converge sequence for some L^(p)(R^(N)).By the properties of the weaker potential function,the compact imbedding theorem on the bound domain along with three special partitions of the entire space,the compact imbedding theorem was verified for some kind of Sobolev space with weaker potential function.And such a theorem could be useful for the study of the existence of a solution for some type of the elliptic equations which possess this kind of potential function when compactness for the functional space fails.