具有指数非线性项的多重调和方程
发布时间：2018-11-07 14:26
【摘要】：近年来,高阶非线性偏微分方程的研究日益受到重视.这是因为此类方程已经被广泛地应用于描述经典力学中的弹性薄板形变模型、稳态的曲面扩散流模型、生物物理学中的Hilfrich模型、微分几何中的Willmore曲面及Paneitz-Branson方程中的各种丰富现象,具有强烈的实际背景；另一方面,从数学层面上说,在高阶方程的研究中,对数学也提出了许多挑战性问题,并且出现了一些新数学现象；此外,在研究中,还可以很好地综合应用偏微分方程基本理论、变分学、非线性分析、几何分析及数学物理等学科的理论知识,进而解决具体的科学问题.本文将重点研究指数增长型多重调和偏微分方程.第一部分研究双调和方程全空间解的径向对称性.在RN中考虑双调和方程△2u=8(N-2)(N-4)eu,其中N≥5.给出此方程的全空间解是径向对称解的充分条件.这一结果丰富了高阶偏微分方程解集的结构和性质,在研究解的几何形态方面,具有一定的理论参考价值.第二部分将重点研究高阶共形不变方程解的存在性.探究R2m中的多重调和方程△mu=土eu,其中m≥2.特别地,我们给出了对任意的V0,方程△mu=-eu存在径向对称解u并使得这表明高阶共形不变方程(-△)mu=Qe2mu,当m是奇数时,任给Q00和任意的V0,在R2m中存在共形度量gu使得Qg。=Q0并且volgu(R2m)=V.第三部分主要研究一类非线性椭圆偏微分方程稳定解的分类问题.首先探究RN中具有指数非线性项的多重调和椭圆方程(-△)mu=eu,其中N2m,m≥3稳定解的存在性问题.我们证明此方程存在许多径向及非径向对称的稳定解,同时结论也表明相比较m≤2时会产生丰富的新现象.其次考虑具有奇异或退化性的散度型椭圆方程-div(|x|a%絬)=|x|reu其中α,γ∈R满足N+α2,γ-α-2,在全空间RN中我们给出了相应的Liouville型结果.
[Abstract]:In recent years, more and more attention has been paid to the study of higher order nonlinear partial differential equations. This is because such equations have been widely used to describe the elastic thin plate deformation model in classical mechanics, the steady surface diffusion flow model, and the Hilfrich model in biophysics. Willmore surfaces in differential geometry and various rich phenomena in Paneitz-Branson equations have strong practical background. On the other hand, from the mathematical level, in the research of higher order equations, many challenging problems have been put forward to mathematics, and some new mathematical phenomena have appeared. In addition, the basic theory of partial differential equation, variational theory, nonlinear analysis, geometric analysis and mathematical physics can be well applied to solve the specific scientific problems. In this paper, we will focus on the exponential growth polyharmonic partial differential equations. In the first part, the radial symmetry of the total space solution of the biharmonic equation is studied. The biharmonic equation 2u=8 (N-2) (N-4) eu, is considered in RN where N 鈮，
本文编号：2316638
[Abstract]:In recent years, more and more attention has been paid to the study of higher order nonlinear partial differential equations. This is because such equations have been widely used to describe the elastic thin plate deformation model in classical mechanics, the steady surface diffusion flow model, and the Hilfrich model in biophysics. Willmore surfaces in differential geometry and various rich phenomena in Paneitz-Branson equations have strong practical background. On the other hand, from the mathematical level, in the research of higher order equations, many challenging problems have been put forward to mathematics, and some new mathematical phenomena have appeared. In addition, the basic theory of partial differential equation, variational theory, nonlinear analysis, geometric analysis and mathematical physics can be well applied to solve the specific scientific problems. In this paper, we will focus on the exponential growth polyharmonic partial differential equations. In the first part, the radial symmetry of the total space solution of the biharmonic equation is studied. The biharmonic equation 2u=8 (N-2) (N-4) eu, is considered in RN where N 鈮，
本文编号：2316638
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