一类积分不等式及其变分计算

王 贝, 雷雨田

(1. 江苏教育学院 数学系, 南京 210013; 2. 南京师范大学 数学科学学院, 南京 210046)

经典的Hardy-Littlewood-Sobolev(HLS)不等式为[1]

(1)

1) 加权HLS不等式[2]：

(2)

2) Wolff型不等式[3]：

(3)

其中:Wβ,γ(f)是正可积函数f的Wolff位势：

Iα(f)是f的Riesz位势：

为研究式(2)的最佳常数, Lieb[4]考虑了泛函

在约束条件‖f‖r=‖g‖s=1下的极大化问题, 并证明了极大元是径向对称且单调下降的. 此时, Euler-Lagrange方程组为

(4)

Jin等[5]利用积分形式的移动平面法, 证明了方程组的正解是对称单调的; 随后, 他们利用关于正则性的lifting引理, 得到了正解的可积性[6]. 基于此, 文献[7-9]计算了正解的渐近估计.

特别地, 当α=β=0时, 式(4)退化为HLS不等式最佳常数问题对应的Euler-Lagrange方程组:

(5)

(6)

当α=2时, 式(6)即为Lane-Emden方程组:

(7)

其正解的存在性问题即为Lane-Emden猜想[10].

作为含有Riesz位势方程组(5)的自然推广, 考虑含有Wolff位势的方程组:

(8)

文献[11-12]分别得到了其正解的可积性和对称单调性. 在此基础上, 文献[13]得到了正解当x→∞时的衰减估计; 文献[14]将对称性和衰减估计推广到多个方程联立的方程组.

利用文献[15]的结果及衰减估计可以研究γ-Laplace方程正解的渐近行为[16-17], 利用文献[18]的结果可以研究k-Hessian方程解的整体性质.

本文结合HLS不等式和Wolff不等式, 给出一种新的Wolff型位势的积分估计, 并给出了比式(4)更一般的变分结果.

1 积分不等式

定理1设g≥0,g∈Lt(Rn). 记G(x)=Wβ,γ(g)(x), 则G∈Ls(Rn), 且

(9)

(10)

证明: 注意到HLS不等式(1)蕴含

‖Iβγ(g)‖p≤C‖g‖np/(n+pβγ).

此即式(9).

反之, 利用式(10)可知

证毕.

2 Euler-Lagrange方程

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

且不等式E(f,g)≤C‖f‖r‖g‖s存在最佳常数C*, 则对应的最佳函数f,g满足:

(11)

经计算, 得

(12)

注意到E(f*,g*)=C*, 式(12)即为

类似地, 可得

运用变分法基本原理, 可得式(11). 证毕.

(13)

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