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Hardy–Littlewood–Sobolev inequality and existence of the extremal functions with extended kernel

Part of: Nonlinear integral equations Partial differential equations

Published online by Cambridge University Press:  31 October 2022

Zhao Liu
Zhao Liu*
Affiliation:
School of Mathematics and Computer Science, Jiangxi Science and Technology Normal University, Nanchang 330038, P. R. China (liuzhao@mail.bnu.edu.cn)
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Abstract

In this paper, we consider the following Hardy–Littlewood–Sobolev inequality with extended kernel(0.1)

\begin{equation} \int_{\mathbb{R}_+^{n}}\int_{\partial\mathbb{R}^{n}_+} \frac{x_n^{\beta}}{|x-y|^{n-\alpha}}f(y)g(x) {\rm d}y{\rm d}x\leq C_{n,\alpha,\beta,p}\|f\|_{L^{p}(\partial\mathbb{R}_+^{n})} \|g\|_{L^{q'}(\mathbb{R}_+^{n})}, \end{equation}
for any nonnegative functions $f\in L^{p}(\partial \mathbb {R}_+^{n})$, $g\in L^{q'}(\mathbb {R}_+^{n})$ and $p,\,\ q'\in (1,\,\infty )$, $\beta \geq 0$, $\alpha +\beta >1$ such that $\frac {n-1}{n}\frac {1}{p}+\frac {1}{q'}-\frac {\alpha +\beta -1}{n}=1$.

We prove the existence of all extremal functions for (0.1). We show that if $f$ and $g$ are extremal functions for (0.1) then both of $f$ and $g$ are radially decreasing. Moreover, we apply the regularity lifting method to obtain the smoothness of extremal functions. Finally, we derive the sufficient and necessary condition of the existence of any nonnegative nontrivial solutions for the Euler–Lagrange equations by using Pohozaev identity.

Keywords

Existence of extremal functionsEuler–Lagrange equationsPohozaev identityHardy–Littlewood–Sobolev inequality

MSC classification

Primary: 45G15: Systems of nonlinear integral equations
Secondary: 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 5 , October 2023 , pp. 1683 - 1705
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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