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 dxvvCvvax 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 )(sup 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 positive solutions to a class equations with a singular coefficien and critical exponent.Adv. Differential Equations,1996,2：241-264 3 Jannelli E.The role played by space dimension in elliptic problems.J.Differential Equations,1999,156：407-426 4 Ferrero A,Gazzola F.Existence of solutions for singular critical growth semilinear elliptic equations.J.Differential Equations, 2001,177：494-522 5 Cao D,Peng S.A note on the sign-changing solutions to elliptic problems with critical Sobole and Hardy terms.J.Differential Equations,2003,193：424-434 6 Kang D,Peng.S.The existence of positive solutions for elliptic equations with critical Sobolev-Hardy exponents.Appl Math Letters,2004,17：411-416 7 Smets D.Nonlinear Schrodinger equations with Hardy potiential and critical nonlinearities.Trans Amer Math Soc,2005,357：2909-2938 8 Cao D,Han H.Solutions to critical elliptic equations with Multi-singular inverse square potentials.J.Differential Equations, 2006,224：332-372 9 kang D.On the elliptic problems involving multi-singular inverse square potentials and critical Sobolev exponent. Nonlinear Analysis, 2008,68：3050-3066 10 安智，康东升一类奇异椭圆问题的解的渐近性中南民族大学学报，2009， 28(1)：116-119 11 Kang D,Shuang J.Solutions for semilinear elliptic problems with critical Sololev-Hardy exponents and Hardy potential.Applied Mathematics Letters,2005,18：1094-1100 12 Kang D,Deng Y.Multiple solutions for inhomogeneous ellipic problems involving critical Sobolev-Hardy exponents. Nonlinear Analysis,2005,60：729-753 13 Brezis H,Kato T.Remarks on the Schrodinger operator with singular complex potentials.Math.Pures Appl,1979,58：137-151 14 Cao D,Han P.Solutions for semilinear elliptic equations with critical exponents and Hardy potential.J.Differential Equations,2004,205：521-537 15 Chou K,Chu C.On the best constant for a weighted Sobolev-Hardy inequality.London Math.Soc,1993,48：137-151 一类带有临界 Sobolev 指数的半线性椭圆方程的研究 23 16 Felli V,Terrcini S.Ellipic equations with multi-singular inverse-square potentials and critical nonlinearity.Partial Differential Equations,2006,31：469-495 17 Gao W,Peng S.An elliptic equation with combined critical Sobolev-Hardy terms.Nonlinear Analysis,2006,65：1595-1612 18 Han P.Quasilinear elliptic problems with critical exponents and Hardy terms. Nonlinear Analysis,2005,61：737-758 19 Kang D,Peng S.Existence of solutions for elliptic equations with critical Sobolev-Hardy exponents.Nonlinear Analysis,2004,56：1151-1164 20 Peng S.Remarks on singular critical growth elliptic equations.Discrete Contin .Dyn.Syst,2006,14：707-719 21 Brezis H,Nirenberg L.Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent.Pure Appl.Math,1983,36：437-477 22 Kang D,Deng Y.Existence of solution for a singular critical elliptic equation.Math. Anal.Applied,2003,284：724-732 23 Kang D,Peng.S.Existence of solutions for elliptic problems with critical Sobolev-Hardy exponents.Israel J.Math,2004,143 24 Athkinson F V,Brezis H,Peletier L A.Nodal solutions of elliptic equations with critical Sobolev exponent.J.Differential Equations,1990,5:151-170 25 Catrina L,Wang Z.On the Caffarelli-Kohn-Nirenberg inqualities:sharp constants,existence(and nonexistence),and symmetry of extremal functions.Comm.Pure Appl.Math,2001,54:229-257 26 Ekeland L,Ghoussoub N.Selected new aspects of the calculus of variations in the large.Bull.Amer.Math,Soc.,2002,39:207-265 27 Ferrero A,Gazzola F.Existence of solutions for singular critical growth semilinear elliptic equations. J.Differential Equations,2001,177:494-522 28 Garcia Azorero J P,Peral Alonso I.Hardy inequalities and some critical and parabolic problems. J.Differential Equations,1998,144:441-476 29 Ghoussoub N,Yuan C.Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents.Trans.Amer.Math.Soc.,2000,352: 5703-5743 30 Jannelli E.The role played by space dimension in elliptic critical problems. J