本文選題：預(yù)定曲率方程　切入點(diǎn)：能量估計(jì)　出處：《華東師范大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：預(yù)定曲率問(wèn)題是黎曼幾何中的經(jīng)典問(wèn)題,對(duì)于一個(gè)黎曼流形M,以及光滑函數(shù).f,通過(guò)求解M對(duì)應(yīng)的預(yù)定曲率方程可以找到M上的一個(gè)度量,使得M在新的度量下的曲率為f,并且新的度量與M上的標(biāo)準(zhǔn)度量是共形的.本文研究的是預(yù)定曲率類型的方程-△u + u = Q(|x|)u~p,u＞O,u∈H1(R),(0.1)其中Q(r)是一 個(gè)正函數(shù);當(dāng)N≥3時(shí),1＜p＜(?);當(dāng)N = 2時(shí),1＜p＜ +∞.本文將證明,當(dāng)r→ +∞時(shí),若Q(r)有如下展開(kāi)式(?),其中α,m,,θ,Q0是常數(shù),且α＞0,m＞1,θ＞0,Q0＞0,則方程(0.1)有無(wú)窮多個(gè)非徑向正解,并且解的能量可以任意大.
[Abstract]:The problem of predetermined curvature is a classical problem in Riemannian geometry. For a Riemannian manifold M and smooth function .f, a metric on M can be found by solving the equation of predetermined curvature corresponding to M. Let M's curvature be f under the new metric, and the new metric is conformal with the standard metric on M. In this paper, we study the equation of predefined curvature type -u = Q (x ~ u ~ u ~ PU > O _ u 鈭，

本文編號(hào)：1622450