一類積分不等式及其變分計(jì)算

王 貝, 雷雨田

(1. 江蘇教育學(xué)院 數(shù)學(xué)系, 南京 210013; 2. 南京師范大學(xué) 數(shù)學(xué)科學(xué)學(xué)院, 南京 210046)

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

(1)

1) 加權(quán)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(shù)f的Wolff位勢(shì)：

Iα(f)是f的Riesz位勢(shì)：

為研究式(2)的最佳常數(shù), Lieb[4]考慮了泛函

在約束條件‖f‖r=‖g‖s=1下的極大化問(wèn)題, 并證明了極大元是徑向?qū)ΨQ且單調(diào)下降的. 此時(shí), Euler-Lagrange方程組為

(4)

Jin等[5]利用積分形式的移動(dòng)平面法, 證明了方程組的正解是對(duì)稱單調(diào)的; 隨后, 他們利用關(guān)于正則性的lifting引理, 得到了正解的可積性[6]. 基于此, 文獻(xiàn)[7-9]計(jì)算了正解的漸近估計(jì).

特別地, 當(dāng)α=β=0時(shí), 式(4)退化為HLS不等式最佳常數(shù)問(wèn)題對(duì)應(yīng)的Euler-Lagrange方程組:

(5)

(6)

當(dāng)α=2時(shí), 式(6)即為L(zhǎng)ane-Emden方程組:

(7)

其正解的存在性問(wèn)題即為L(zhǎng)ane-Emden猜想[10].

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

(8)

文獻(xiàn)[11-12]分別得到了其正解的可積性和對(duì)稱單調(diào)性. 在此基礎(chǔ)上, 文獻(xiàn)[13]得到了正解當(dāng)x→∞時(shí)的衰減估計(jì); 文獻(xiàn)[14]將對(duì)稱性和衰減估計(jì)推廣到多個(gè)方程聯(lián)立的方程組.

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

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

1 積分不等式

定理1設(shè)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(yùn)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(shè)f,g≥0,f∈Lr(Rn),g∈Ls(Rn). 如果函數(shù)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(shù)C*, 則對(duì)應(yīng)的最佳函數(shù)f,g滿足:

(11)

經(jīng)計(jì)算, 得

(12)

注意到E(f*,g*)=C*, 式(12)即為

類似地, 可得

運(yùn)用變分法基本原理, 可得式(11). 證畢.

(13)

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