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1、中南民族大学 硕士学位论文 一类带有临界Sobolev指数的半线性椭圆方程的研究 姓名：安智 申请学位级别：硕士 专业：应用数学 指导教师：康东升 20100524 一类带有临界 Sobolev 指数的半线性椭圆方程的研究 I 摘摘 要要 本文主要研究如下半线性椭圆方程： = += = 。xu xuuuu ax u u q k i i , 0 , 222 1 2 * 这里)3(NR N 是一光滑有界区域，),2, 1( ,kiai=各不相 等，2k，R， * 22，当足够小的时候，存在正常数 2 C及 3 C，且 32 CC 有 vdxvvax i s mii i )1(2 2 2 dxvvC
2、vvax s mi s mii i )( )1(22 22 )1(22 2 + （2.2.4） 当 * 22,由 Young 不等式有 )( )1(2 2 i aB i s mi udxuu 中南民族大学硕士学位论文 14 + )( 2 )1(22 )( 2 )1(22 )( ii aB i s m aB s mi dxuuCdxuu 。 （2.3.3） 实际上， * 22，则可推出 1)22( * t， * 2 1 2 2 和0C，使得： dxvCOS N tvI t NN + 2 2 2 2 )( 1 )( 0 sup 引理 3.1.4 【18】假设 N N ，也可得到 2 1 1 )(s
3、up N Aw S N cwI+ 证毕。 注：注：这里只给出了当方程()1 . 1 . 1中2=q 时的证明过程，当 * 22 q时的证明相 对来说比较复杂，因此，在这种情况下，我们用本文中的方法不太容易得出方程 (1.1.1) 的变号解的存在性这有待于新的思路和方法来研究。 中南民族大学硕士学位论文 22 参考文献参考文献 1 Caffarelli L, Kohn R, Nirenberg L.First order interpolation inequalit with weights.Compositio Math,1984,53：259-275 2 Terracini S.On po
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