1、 收稿日期: 2000- 03- 09* 基金项目:SE1 S “(19871030); 0, x I 85u5n = 0, x I 5 8(1)T: E 0 Bl ; TN 3,52 0, PEI (0, E0 H,Z(1)iuE,uE1USM f , OuEB Kv.1 记号和定义 1 -, 5Bt:|. !H1e( 8) H1( 8)B0 bW,H1e = f1M , !U(x) / 5B3:- $u + u = up-1, x I RNu 0, x I RNu(0) = maxxI RNu(x)u I H1( RN),D4 U(x) | x|f , O1| x |/, lim|x| y
2、+ U(x)e|x| xN-12 = c0 0.TMy = xE, !8E= y: Ey I 8,5 T(1)N- $u + u = Q(Ey)up-1, y I 8Eu 0, y I 8E5u5n = 0 y I 5 8E(2) 2 v( 自然科学版)29 2 Journal of South China Universityof Technology Vol.29 No.22001 M2(NaturalScienceEdition) Februany 2001 B, !Q(0) = 1,C p T(2) / T:uE= U(y) + vE(y) (3)T,vE(y) I H1e( 8E),
3、 OvE = Q8EDvE 2+ v2E y0(E y 0 H). D12 , T(2) T T(3),5uEB Kv.lIE(u) = 12Q8EDu 2 + u2 - 1pQ8Eu p,u I H1( 8E),5u T(2) Nu IE(u) “.Z5 ,N 5.2 定理的证明:JE(v) = IE(U+ v), Pv I H1e( 8E) (4)5 A, Ov JE(v) “ H,u =U+ v IE(u) “. D5,|JE(v)v = 0Z 7 /:JE(v) = JE(0) + fE(v)+ 12QE(v)+ RE(v) (5)TfE(v) = 3U,v4- Q8EQ(Ey)Up-
4、1v,QE(v) = v 2- (p - 1)Q8EQ(Ey)Up- 1v2.7RE(v) R(i)E (v) = O( v min(p-i,3- i), i = 0,1,2. R(i)E RE(v)iQs.RietzVC V,if*E I H1e( 8E), P3f*E ,v4= fE(v), Pv I H1e( 8E).lQ*E(u,v) = - (p- 1)Q8EQ(Ey)Up-2uv,Pu,v I H1e( 8E),RietzVC , iH1e( 8E)H1e( 8E)L LE, PQ*E(u,v) = ,V7QE(v) = ,# T(5) VJE(v) = JE(0)+ 12 +RE
5、(v),#JcE(v) = f*E + LEv+ RcE(v) (6)7F(f,v) = f+ LEv+ RcE(v), f,v I H1e( 8E)(7)5F(0,0) = 0, 5F5v(0,0) = LE.V7: TLE:H1e( 8E) y H1e( 8E) V I, OL-1E c(E 01),5 f ,iD 0(E 01), P if I H1e( 8E)f 0, PEI(0,E0 H,. f*E I1+ I2 (8)6BZ ,I1 Q5 8E5U5n2 12 Q5 8Ev 212 c Q5 8E5U5n2 12v cexp - 12 d(0,58)E v (9)I2 c Q8E1
6、- Q(Ey) p/(p- 1)Up1- 1pv cEQ8Ey p/(p-1)Up1- 1pv cE v (10)5 T(8) (10)fEv c exp - 12 d(0,5 8)E + E v ,V7: TE 0 O sl,5f *E c exp - 12 d(0,58)E + E D.(2)2 2 v(自然科学版)29 ic0 0, PLEv c0 v .QE. !iEi y 0,vi I H1e( 8Ei), PLEivi = O 1i U y 0, (i y+ )Q8EiDviDU+ viU- (p- 1)Q8EQ(Eiy)Up-2viUy 0(11)vi = 1, V !vi _v
7、 I H1e(RN),vi y vC2loc( RN),%U I C 0 ( RN), U1M f , 7i y+ ,5 T(11) |KQRNDvDU+ vU- (p - 1)QRNUp-2vU= 0 (12)v1M f ,# T(12)v - $v+ v = (p - 1)Up-1v, y I RN.D2, v = 6Ni= 1ai 5Ui5xi x= 0. (13)T,Ux(y) = U(y- x),v f ,7 T(13)|Hai = 0(i = 1,2, ,N) f ,#a1= ,= aN = 0,v = 0,#v y 0,C2loc(RN),V7Q8EUp-2v2i = Q8EHB
8、R(0)Up-1v2i + Q8c H|y| RUp-1v2i y0,Q8EiDvi 2+ v2i - (p- 1)Q8EiUp-2v2i = O 1i y0,#vi 2 = Q8EiDvi 2+ v2i = (p - 1)Q8EiUp-1v2i +O 1i y 0,vi = 1 .y7LEv c0 v ,NL-1E 1c0. 8.参考文献:1 Ni W M,Takagi I. On the shape of the least energysolution to a semilinear Neumann problem J.Comm Pure Appl Math,1991,41:819- 8
9、51.2 Ni W M,TakagiI.Locating the peaksof least energysolutionsto a semilinear Neumann problem J.DukeMath J,1993,70:247- 281.3 Knong M K.Uniqueness of positive solutions of - $u+u= up in RN J.Arch Rat Mech Anal, 1989, 105:243- 266.4 Giadas B, Ni W M,Nirenberg L.Symmetry of pos-itive solutions of nonl
10、ine elliptic equations in RnA.Mathematical Analysis and Applications M. NewYork:Academic press,1981.369- 402.5 Rey O.The role of the Green. sfunctionsin a nonlin-ear elliptic equation involving the critical Sobolev ex-ponent J.JFunct Anal,1990,89:1- 52.Symmetric Solution for a NeumannProblemYao Yang
11、_xin1 Xu Jin_quan2(1. Dept. of Applied Mathematics, South China Univ. of Tech., Guangzhou 510640, China;(2. Dept. of Mathematics, HuizhouUniversity, Huizhou516015, China)Abstract: In this paper, symmetricsolution with exactly one local maximum point is constructed for asingularly perturbed Neumann problem.Key words: singularly perturbed; Neumann problem; symmetricsolution2 -: Neumann53
