一類積分不等式及其變分計算

王 貝, 雷雨田

(1. 江蘇教育學院 數學系, 南京 210013; 2. 南京師范大學 數學科學學院, 南京 210046)

經典的Hardy-Littlewood-Sobolev(HLS)不等式為[1]

(1)

1) 加權HLS不等式[2]：

(2)

2) Wolff型不等式[3]：

(3)

其中:Wβ,γ(f)是正可積函數f的Wolff位勢：

Iα(f)是f的Riesz位勢：

為研究式(2)的最佳常數, Lieb[4]考慮了泛函

在約束條件‖f‖r=‖g‖s=1下的極大化問題, 并證明了極大元是徑向對稱且單調下降的. 此時, Euler-Lagrange方程組為

(4)

Jin等[5]利用積分形式的移動平面法, 證明了方程組的正解是對稱單調的; 隨后, 他們利用關于正則性的lifting引理, 得到了正解的可積性[6]. 基于此, 文獻[7-9]計算了正解的漸近估計.

特別地, 當α=β=0時, 式(4)退化為HLS不等式最佳常數問題對應的Euler-Lagrange方程組:

(5)

(6)

當α=2時, 式(6)即為Lane-Emden方程組:

(7)

其正解的存在性問題即為Lane-Emden猜想[10].

作為含有Riesz位勢方程組(5)的自然推廣, 考慮含有Wolff位勢的方程組:

(8)

文獻[11-12]分別得到了其正解的可積性和對稱單調性. 在此基礎上, 文獻[13]得到了正解當x→∞時的衰減估計; 文獻[14]將對稱性和衰減估計推廣到多個方程聯立的方程組.

利用文獻[15]的結果及衰減估計可以研究γ-Laplace方程正解的漸近行為[16-17], 利用文獻[18]的結果可以研究k-Hessian方程解的整體性質.

本文結合HLS不等式和Wolff不等式, 給出一種新的Wolff型位勢的積分估計, 并給出了比式(4)更一般的變分結果.

1 積分不等式

定理1設g≥0,g∈Lt(Rn). 記G(x)=Wβ,γ(g)(x), 則G∈Ls(Rn), 且

(9)

(10)

證明: 注意到HLS不等式(1)蘊含

‖Iβγ(g)‖p≤C‖g‖np/(n+pβγ).

此即式(9).

反之, 利用式(10)可知

證畢.

2 Euler-Lagrange方程

定理2設f,g≥0,f∈Lr(Rn),g∈Ls(Rn). 如果函數K(x,y)>0, 使得如下泛函有意義

且不等式E(f,g)≤C‖f‖r‖g‖s存在最佳常數C*, 則對應的最佳函數f,g滿足:

(11)

經計算, 得

(12)

注意到E(f*,g*)=C*, 式(12)即為

類似地, 可得

運用變分法基本原理, 可得式(11). 證畢.

(13)

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