【摘要】：近年來(lái),高階非線性偏微分方程的研究日益受到重視.這是因?yàn)榇祟?lèi)方程已經(jīng)被廣泛地應(yīng)用于描述經(jīng)典力學(xué)中的彈性薄板形變模型、穩(wěn)態(tài)的曲面擴(kuò)散流模型、生物物理學(xué)中的Hilfrich模型、微分幾何中的Willmore曲面及Paneitz-Branson方程中的各種豐富現(xiàn)象,具有強(qiáng)烈的實(shí)際背景；另一方面,從數(shù)學(xué)層面上說(shuō),在高階方程的研究中,對(duì)數(shù)學(xué)也提出了許多挑戰(zhàn)性問(wèn)題,并且出現(xiàn)了一些新數(shù)學(xué)現(xiàn)象；此外,在研究中,還可以很好地綜合應(yīng)用偏微分方程基本理論、變分學(xué)、非線性分析、幾何分析及數(shù)學(xué)物理等學(xué)科的理論知識(shí),進(jìn)而解決具體的科學(xué)問(wèn)題.本文將重點(diǎn)研究指數(shù)增長(zhǎng)型多重調(diào)和偏微分方程.第一部分研究雙調(diào)和方程全空間解的徑向?qū)ΨQ(chēng)性.在RN中考慮雙調(diào)和方程△2u=8(N-2)(N-4)eu,其中N≥5.給出此方程的全空間解是徑向?qū)ΨQ(chēng)解的充分條件.這一結(jié)果豐富了高階偏微分方程解集的結(jié)構(gòu)和性質(zhì),在研究解的幾何形態(tài)方面,具有一定的理論參考價(jià)值.第二部分將重點(diǎn)研究高階共形不變方程解的存在性.探究R2m中的多重調(diào)和方程△mu=土eu,其中m≥2.特別地,我們給出了對(duì)任意的V0,方程△mu=-eu存在徑向?qū)ΨQ(chēng)解u并使得這表明高階共形不變方程(-△)mu=Qe2mu,當(dāng)m是奇數(shù)時(shí),任給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類(lèi)非線性橢圓偏微分方程穩(wěn)定解的分類(lèi)問(wèn)題.首先探究RN中具有指數(shù)非線性項(xiàng)的多重調(diào)和橢圓方程(-△)mu=eu,其中N2m,m≥3穩(wěn)定解的存在性問(wèn)題.我們證明此方程存在許多徑向及非徑向?qū)ΨQ(chēng)的穩(wěn)定解,同時(shí)結(jié)論也表明相比較m≤2時(shí)會(huì)產(chǎn)生豐富的新現(xiàn)象.其次考慮具有奇異或退化性的散度型橢圓方程-div(|x|a%絬)=|x|reu其中α,γ∈R滿(mǎn)足N+α2,γ-α-2,在全空間RN中我們給出了相應(yīng)的Liouville型結(jié)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，

本文編號(hào)：2316638