关于一类半线性椭圆方程的研究-应用数学专业论文.docxVIP

⑨ ⑨ 博士学位论文 ⅨX小报^LⅨ豁彤巩盯10N 关簟illt半线性椭西方程，径向对称解，正解(正则解)，(正)奇解。单调分层性，渐近行 为。奇性，存在性 ⑨ ⑨ 博士学位论文 DO∞10R^L Di$$EJzrATION ABSTRACT This dissertation investigates a class of semilinear elliptic equations： △让+￡冬l龟Iz产t一=0 in r， (0．1．1) div(A([x矽“)+K(I*I)uP=0 in r (o．1．2) and Au+KCIzl)u+M(Izl)=0 in戤， (0．1．3) where n≥3，△=∑冬1筹is the n-dimension Laplacian operator，P1 and pi1，岛--2，c‘0，i=1，2 ，七．A(Izl)0，K(M)柚d，(k1)are HSlder continuous nonnegative functions． The equations arise in Riemannian geometry,are said to be the contformal scalar curvature equations．The first equation after simplication is Au+K(Izl)u鹈=0． For the physical reasons and because of the results on the symmetry of positive solutions，most of mathematicians study the radial solutions of them，such as the existence of positive radial solutions and asymptotic behavior at infinity etc． The underlying obstacles result from the presence of the second terms(K(IzI) or c‘忙h i=1，2，一·，毛inhomogeneity and the lack of compactness due to the unbounded domain(entire space)．This dissertation deserve to the radial solutions， with r=kl’the clasB of equations reduce to the following respectively t工，，+!三!∥+∑叁lQ—t一=o，r2 o， (o。1．4) (r“一1A(r)∥)’+rn-1K(r)t，=0， r 2 0 (0．1．5) and t，+竺．}∥+Ⅳ(r)t‘，+p，(r)=o， r2 o． (o．1．6) Denote the positive radial solutions of the class of equatons‰(r)with initial value t‘(o)=OL For every口0，there is a positive solution‰(r)for Eq．(0．1．4)and (0．1，5)祈th Some assumptions on A，屯，1 S i≤k and^(p1)，K(I*I)respectively． But there is only multiple many positive entire solutions for Eq．(0．1．6)．This dis- sertation deserve to study the existence of positive solutions respective to the initial ⑨ ⑨ 博士学位论文 D 0lCTOR^L DISSERT^110N value u(O)=o0 of Eq．(o．1．6)and the monotonicity property with respective to the initial values of positive radial solutions of above three equations．and their asymptotic behaviors at infinity respectively．Meanwhile，we get some results about