Xing-Liang Tian

On the Stein-Weiss inequalities and higher-order Caffarelli-Kohn-Nirenberg type inequalities: sharp constants, symmetry of extremal functions

Shengbing Deng, Xingliang Tian

In this paper, we first classify all radially symmetry solutions of the following weighted fourth-order equation \begin{equation*} Δ(|x|^{-γ}Δu)=|x|^γu^{\frac{N+4+3γ}{N-4-γ}},\quad…

Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities: the singular case

Shengbing Deng, Xingliang Tian

Let us consider the following Caffarelli-Kohn-Nirenberg type inequality \begin{equation}\label{nsckn} \int_{\mathbb{R}^N}|x|^{-β}|\mathrm{div} (|x|^α\nabla u)|^2 \mathrm{d}x \geq \…

Gradient stability of Caffarelli-Kohn-Nirenberg inequality involving weighted p-Laplace

Shengbing Deng, Xingliang Tian

The best constant and extremal functions are well known of the following Caffarelli-Kohn-Nirenberg inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p\frac{\mathrm{d}x}{|x|^μ}\geq \mathc…

A note on the Lp-Sobolev inequality

Shengbing Deng, Xingliang Tian

The usual Sobolev inequality in $R^{N}$ , asserts that $∥\nabla u ∥_{L^{p} (R^{N})} \geq S ∥ u ∥_{L^{p^{*}} (R^{N})}$ for $1 < p < N$ and $p^{*} = \frac{pN}{N - p}$ ,…

Symmetry breaking of extremals for the high order Caffarelli-Kohn-Nirenberg type inequalities

Shengbing Deng, Xingliang Tian

In this paper we give the first result about the precise symmetry and symmetry breaking regions of extremal functions for weighted second-order inequalities. Firstly, based on the…

Classification and non-degeneracy of positive radial solutions for a weighted fourth-order equation and its application

Shengbing Deng, Xingliang Tian

This paper is devoted to radial solutions of the following weighted fourth-order equation \begin{equation*} \mathrm{div}(|x|^α\nabla(\mathrm{div}(|x|^α\nabla u)))=u^{2^{**}_α-1},\q…

On the stability constant of Caffarelli-Kohn-Nirenberg inequality

Shengbing Deng, Xingliang Tian

By using a spectral analysis, we first show that the Caffarelli--Kohn--Nirenberg inequality with gradient remainder term of any order less than $4$ does not hold on the {\em Felli-…

Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian

Shengbing Deng, Xingliang Tian

For $N \geq 5$ and $0 < μ < N - 4$ , we first show a non-degenerate result of the extremal functions for the following Rellich-Sobolev type inequality \begin{align*} \int_{\mathbb{R}^N}|Δu…

Remainder terms of a nonlocal Sobolev inequality1

Shengbing Deng, Xingliang Tian, Minbo Yang +1

In this note we study a nonlocal version of the Sobolev inequality \begin{equation*} \int_{\mathbb{R}^N}|\nabla u|^2 dx \geq S_{HLS}\left(\int_{\mathbb{R}^N}\big(|x|^{-α} \ast u^{2…

Stability of Hardy-Sobolev inequality involving p-Laplace

Shengbing Deng, Xingliang Tian

This paper is devoted to considering the following Hardy-Sobolev inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p \mathrm{d}x \geq \mathcal{S}_β\left(\int_{\mathbb{R}^N}\frac{|u|^{p^*…

Caffarelli-Kohn-Nirenberg-type inequalities related to weighted $p$ -Laplace equations

Shengbing Deng, Xingliang Tian

We use a suitable transform related to Sobolev inequality to investigate the sharp constants and optimizers for some Caffarelli-Kohn-Nirenberg-type inequalities which are related t…

Some weighted fourth-order Hardy-Henon equations

Shengbing Deng, Xingliang Tian

By using a suitable transform related to Sobolev inequality, we investigate the sharp constants and optimizers in radial space for the following weighted Caffarelli-Kohn-Nirenberg-…

Boundary concentration of peak solutions for fractional Schrödinger-Poisson system

Shengbing Deng, Xingliang Tian

The goal of this paper is to study the existence of peak solutions for the following fractional Schrödinger-Poisson system: \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{arra…