关于非线性多调和方程的整体解的存在性

Existence of Entire Solutions to Nonlinear Poly-harmonic Equations in R^2

本文主要研究形如：△ ((△nu)p ) + f(|x|, u, ?u)=0, x∈R2的非线性多调和方程的整体解， ?1*此处 n 是自然数，p>1 是实常数，f: R+ × R× R → R+是一个连续函数, ξα*:=|ξ|α?1ξ，ξ∈R，α>0，证明了该方程不存在径向对称的正整体解, 并给出存在无穷多个最终为负值且其渐进阶（当 n→∞时，|u| 作为无穷大量的阶）不低于 |x|2 log|x| 的整体解 u 的充分条件及渐进阶正好是 |x|2 log|x| 的 n n充分必要条件.

In this paper two-dimensional nonlinear poly-harmonic equations of the form are considered, where p>1, b0, n is an integer (n1), xa*:=|x|a-1x, xR, a>0, and f:+RRR R+ is a continuous function. It is shown that any radially symmetric entire solution grows at least as fast as positive constant multiplies of-|x|2n(log|x|)1/(p-1) as |x|. It is given that some sufficient conditions and necessary conditions for the existence of infinitely many symmetric entire solutions which are asymptotic to positive constant multiples of -|x|2n(log|x|)1/(p-1) as |x|.