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${\mbox{\boldmath $R$}}^N$上奇异非线性多调和方程的正整体解
引用本文:吴炯圻.${\mbox{\boldmath $R$}}^N$上奇异非线性多调和方程的正整体解[J].系统科学与数学,2004,24(3):332-339.
作者姓名:吴炯圻
作者单位:漳州师范学院数学系 福建漳州
基金项目:国家自然科学基金(19871068),福建省自然科学基金(F00018)资助课题.
摘    要:本文研究形如△((△nu)(p-1) )=f(|x|,u,|(?)u|)u-β,x∈RN的奇异非线性多调和方程在RN上的正整体解,此处P>1,β≥0是常数,n是自然数,f:R × R ×R →R 是一个连续函数, ξδ*:=sign(ξ)·|ξ|δ,,ξ∈R,δ>0,给出了该类方程具有无穷多个其渐进阶刚好为|x|2n的正整体解的充分条件与必要条件.这些结论可以推广到更一般的方程.

关 键 词:非线性多调和方程  奇异方程  正整体解  不动点定理
修稿时间:2002年5月8日

ON POSITIVE ENTIRE SOLUTIONS TO SINGULAR, NONLINEAR POLY-HARMONIC EQUATIONS IN ${\mbox{\boldmath $R$}}^N(N \geq 3)$
Jiong Qi WU.ON POSITIVE ENTIRE SOLUTIONS TO SINGULAR, NONLINEAR POLY-HARMONIC EQUATIONS IN ${\mbox{\boldmath $R$}}^N(N \geq 3)$[J].Journal of Systems Science and Mathematical Sciences,2004,24(3):332-339.
Authors:Jiong Qi WU
Institution:Zhangzhou Teachers College, Zhangzhou, fujian 363000
Abstract:In this paper N-dimensional singular, nonlinear poly-harmonic equations of the following form $$ \triangle ((\triangle^n u)^{(p-1)^*})=f(|x|, u, |\triangle u|)u^{-\beta},\q x \in {\mbox{\boldmath $R$}}^N $$ are considered, where $p>1$, $\beta \geq 0$, $n$ is an integer ($n \geq 1$), $\xi^{\delta^*}:={\rm sign}(\xi)\cdot |\xi|^{\delta}$, $\xi \in {\mbox{\boldmath $R$}}$, $\delta>0$. and $f:\overline{R}_+ \times R_+ \times \overline{R}_+ \rightarrow R_+$ is a continuous function. Some sufficient conditions and necessary conditions are obtained for the existence of infinitely many positive symmetric entire solutions which are asymptotic to positive constant multiples of $|x|^{2n}$ as $|x|\rightarrow \infty$. The results can be extended to certain equations of more general form.
Keywords:Non-linear poly-harmonic equation  positive entire solutions  singular equation  fixed point theorem  
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