关于奇异非线性多调和方程的正整体解

On Positive Entire Solutions to Singular Nonlinear Poly-Harmonic Equations in R^2

该文主要研究形如Δ( (Δnu) p- 1*) =f( | x| ,u,| u| ) u-β, x∈ R2的奇异非线性多调和方程在 R2 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 ,ξα*:=| ξ|α- 1ξ,ξ∈R,α>0 .证明了这种解 u必无界且其渐进阶 (当n→∞时 u作为无穷大量的阶 )不低于 | x| 2 nlog| x| ,给出了该方程具有无穷多个其渐进阶刚好为 | x| 2 nlog| x|的正整体解的充分与充分必要条件 .

In this paper, two dimensional singular nonlinear poly harmonic equation of the form Δ((Δ\+nu)\+\{p-1*\}) = f(|x|, u, |u|)u\+\{-β\},\ x∈R\+2 is considered, where p>1, β≥0, n is an integer (n≥1),ξ\+\{α*\}:=|ξ|\+\{α-1\}ξ,ξ∈R,α>0. and f: \-+×R\-+×\-+→R\-+ is a continuous function. It is shown that any positive radially symmetric entire solution grows at least as fast as positive constant multiples 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 positive symmetric entire solutions which are asymptotic to positive constant multiples of |x|\+\{2n\}( log |x|)\+\{1/(p-1)\} as |x|→∞ . The results can be extended to certain equations of more general form, e.g., Δ((Δ\+nu)\+\{p-1*\})=f(|x|, u, |u|,|u\+2u|,\:,|u|\+\{2n-1\})u\+\{-β\},\ x∈R\+2.