arXiv subjects

Xingliang Tian

Publications and source records attributed to Xingliang Tian.

15 recordsLinked to original sources

arXiv2024

On the stability constant of Caffarelli-Kohn-Nirenberg inequality

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-Schneider} curve $b_{\mathrm{FS}}(a)$. Furthermore, we prove the existence of minimizers of sharp stability constant of Caffarelli--Kohn--Nirenberg inequality near the new curve $b^*_{\mathrm{FS}}(a)(>b_{\mathrm{FS}}(a))$, which extends the work of Wei and Wu [Math. Z., 2024] to a sightly larger region.

math.AP↗

arXiv2024

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

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|^2 \mathrm{d}x -C_{μ,1}\int_{\mathbb{R}^N}\frac{|\nabla u|^2}{|x|^2} \mathrm{d}x +C_{μ,2}\int_{\mathbb{R}^N}\frac{u^2}{|x|^4} \mathrm{d}x \geq \mathcal{S}_μ\left(\int_{\mathbb{R}^N}|u|^{\frac{2N}{N-4}} \mathrm{d}x\right)^\frac{N-4}{N},\quad \forall u\in C^\infty_0(\mathbb{R}^N), \end{align*} where $C_{μ,1}$, $C_{μ,2}$ and $\mathcal{S}_μ$ are constants depending on $N$ and $μ$, which is a key ingredient in analyzing the blow-up phenomena of solutions to various elliptic equations on bounded or unbounded domains. Then by using spectral analysis combined with a compactness argument, we consider the stability of this inequality. Furthermore, we derive a remainder term inequality in the weak Lebesgue-norm sense in a subdomain with finite Lebesgue measure.

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arXiv2024

A note on the Lp-Sobolev inequality

The usual Sobolev inequality in $\mathbb{R}^N$, asserts that $\|\nabla u\|_{L^p(\mathbb{R}^N)} \geq \mathcal{S}\|u\|_{L^{p^*}(\mathbb{R}^N)}$ for $1 0$ independent of $Ω$ such that \[ \|\nabla u\|^p_{L^p(Ω)} -\mathcal{S}^p\|u\|^p_{L^{p^*}(Ω)} \geq \mathcal{C}|Ω|^{-\fracγ{p^*(p-1)}} \|u\|_{L^{\bar{p}}_w(Ω)}^γ\| u\|_{L^{p^*}(Ω)}^{p-γ},\quad \mbox{for all}\ u\in C^\infty_0(Ω)\setminus\{0\}, \] where $γ=\max\{2,p\}$, $\bar{p}=p^*(p-1)/p$, and $\|\cdot\|_{L^{\bar{p}}_w(Ω)}$ denotes the weak $L^{\bar{p}}$-norm. Moreover, we establish a sharp upper bound of Sobolev inequality in $\mathbb{R}^N$.

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arXiv2024

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

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 u\geq 0 \quad \mbox{in}\quad \mathbb{R}^N, \end{equation*} where $N\geq 5$, $-2<γ<0$. Then we derive the sharp Stein-Weiss inequality and standard second-order Caffarelli-Kohn-Nirenberg inequality with radially symmetry extremal functions. Moreover, by using standard spherical decomposition, we derive a sharp weighted Rellich-Sobolev inequality. Furthermore, we establish the sharp second-order Caffarelli-Kohn-Nirenberg type inequalities with two variables which have radially symmetry extremal functions. Finally, we derive the weak form Hardy-Rellich inequalities with sharp constants.

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arXiv2024

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

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 work of C.-S. Lin [Comm. Partial Differential Equations, 1986], a new second-order Caffarelli-Kohn-Nirenberg type inequality will be established, i.e., \begin{equation*} \int_{\mathbb{R}^N}|x|^{-β}|\mathrm{div} (|x|^α\nabla u)|^2 \mathrm{d}x \geq \mathcal{S}\left(\int_{\mathbb{R}^N} |x|^β|u|^{p^*_{α,β}} \mathrm{d}x\right)^{\frac{2}{p^*_{α,β}}},\quad \mbox{for all}\ u\in C^\infty_0(\mathbb{R}^N), \end{equation*} for some constant $\mathcal{S}=\mathcal{S}(N,α,β)>0$, where \begin{align*} N\geq 5,\quad α>2-N,\quad α-2<β\leq \frac{N}{N-2}α,\quad p^*_{α,β}=\frac{2(N+β)}{N-4+2α-β}. \end{align*} We obtain a symmetry breaking conclusion: when $α>0$ and $β_{\mathrm{FS}}(α)<β< \frac{N}{N-2}α$ where $β_{\mathrm{FS}}(α):= -N+\sqrt{N^2+α^2+2(N-2)α}$, then the extremal function for the best constant $\mathcal{S}$, if it exists, is nonradial. Furthermore, we give a symmetry result when $β=\frac{N}{N-2}α$ and $2-N<α<0$...

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arXiv2024

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

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 \mathcal{S}\left(\int_{\mathbb{R}^N}|x|^γ |u|^{2^{**}_{α,β}} \mathrm{d}x\right)^{\frac{2}{2^{**}_{α,β}}}, \quad \mbox{for all}\quad u\in C^\infty_0(\mathbb{R}^N\setminus\{0\}), \end{equation} for some $\mathcal{S}=\mathcal{S}(N,α,β)>0$, where $N\geq 5$, $α>2-N$, $\frac{N-4}{N-2}α-4 \leq β\leqα-2$ and \begin{align*} 2^{**}_{α,β}:=\frac{2(N+γ)}{N+2α-β-4} \quad \mbox{with}\quad (N+β)(N+γ)=(N+2α-β-4)^2. \end{align*} A crucial element is that the functional $\int_{\mathbb{R}^N}|x|^{-β}|\mathrm{div} (|x|^α\nabla u)|^2 \mathrm{d}x$ is equivalent to $\int_{\mathbb{R}^N}|x|^{2α-β}|Δu|^2 \mathrm{d}x$. Firstly, we obtain a symmetry result (with partial translation invariant) when $α=0$ and $β=-4$, then existence and non-existence of extremal functions for the best constant $\mathcal{S}$ in \eqref{nsckn} under different conditions are completely given. Moreover, by a result of linearized problem related to radial solution of \eqref{Pwhs0}, we obtain a symmetry breaking conclusion: when $α>0$ and $\frac{N-4}{N-2}α-4<β<β_{\mathrm{FS}}(α)$ where $β_{\mathrm{FS}}(α):= N+2α-4-\sqrt{(N-2+α)^2+4(N-1)}$, the extremal functions for $\mathcal{S}$ are nonradial. Finally, we give a partial symmetry result when $β=\frac{N-4}{N-2}α-4$ and $2-N<α<0$, and we also study the stability of extremal functions.

math.AP↗

arXiv2024

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

The best constant and extremal functions are well known of the following Caffarelli-Kohn-Nirenberg inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p\frac{\mathrm{d}x}{|x|^μ}\geq \mathcal{S} \left(\int_{\mathbb{R}^N}|u|^r\frac{\mathrm{d}x}{|x|^s} \right)^{\frac{p}{r}}, \quad \mbox{for all}\quad u\in C^\infty_c(\mathbb{R}^N), \] where $1<p<p+μ<N$, $\fracμ{p}\leq \frac{s}{r}<\fracμ{p}+1$, $r=\frac{p(N-s)}{N-p-μ}$. An important task is investigating the stability of extremals for this inequality. Firstly, we give the classification to the linearized problem related to the extremals which shows the extremals are non-degenerate. Then we investigate the gradient type remainder term of previous inequality by using spectral estimate combined with a compactness argument which partially extends the work of Wei and Wu [Math. Ann., 2022] to a general $p$-Laplace case, and also the work of Figalli and Zhang [Duke Math. J., 2022] to a weighted case.

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arXiv2023

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

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},\quad u>0\quad \mbox{in}\quad \mathbb{R}^N, \end{equation*} where $N\geq 2$, $\frac{4-N}{2}<α<2$ and $2^{**}_α=\frac{2N}{N-4+2α}$. It is obvious that the solutions of above equation are invariant under the scaling $λ^{\frac{N-4+2α}{2}}u(λx)$ while they are not invariant under translation when $α\neq 0$. We characterize all the solutions to the related linearized problem about radial solutions, and obtain the conclusion of that if $α$ satisfies $(2-α)(2N-2+α)\neq4k(N-2+k)$ for all $k\in\mathbb{N}^+$ the radial solution is non-degenerate, otherwise there exist new solutions to the linearized problem that ``replace'' the ones due to the translations invariance. As applications, firstly we investigate the remainder terms of some inequalities related to above equation. Then when $N\geq 5$ and $0<α<2$, we establish a new type second-order Caffarelli-Kohn-Nirenberg inequality \begin{equation*} \int_{\mathbb{R}^N} |\mathrm{div}(|x|^α\nabla u)|^2 \mathrm{d}x \geq C \left(\int_{\mathbb{R}^N}|u|^{2^{**}_α} \mathrm{d}x\right)^{\frac{2}{2^{**}_α}},\quad \mbox{for all}\quad u\in C^\infty_0(\mathbb{R}^N), \end{equation*} and in this case we consider a prescribed perturbation problem by using Lyapunov-Schmidt reduction.

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arXiv2023

Remainder terms of a nonlocal Sobolev inequality1

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_α^{\ast}}\big)u^{2_α^{\ast}} dx\right)^{\frac{1}{2_α^{\ast}}}, \quad \forall u\in \mathcal{D}^{1,2}(\mathbb{R}^N), \end{equation*} where $S_{HLS}$ is the best constant, $\ast$ denotes the standard convolution and $\mathcal{D}^{1,2}(\mathbb{R}^N)$ denotes the classical Sobolev space with respect to the norm $\|u\|_{\mathcal{D}^{1,2}(\mathbb{R}^N)}=\|\nabla u\|_{L^2(\mathbb{R}^N)}$. By using the nondegeneracy property of the extremal functions, we prove that the existence of the gradient type remainder term and a reminder term in the weak $L^{\frac{N}{N-2}}$-norm of above inequality for all $0<α<N$.

math.AP↗

arXiv2023

Stability of Hardy-Sobolev inequality involving p-Laplace

This paper is devoted to considering the following Hardy-Sobolev inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p \mathrm{d}x \geq \mathcal{S}_β\left(\int_{\mathbb{R}^N}\frac{|u|^{p^*_β}}{|x|^β} \mathrm{d}x\right)^\frac{p}{p^*_β},\quad \forall u\in C^\infty_0(\mathbb{R}^N), \] for some constant $\mathcal{S}_β>0$, where $1<p<N$, $0\leq β<p$, $p^*_β=\frac{p(N-β)}{N-p}$. Firstly, since this problem involves quasilinear operator, we need to establish a compact embedding theorem for some suitable weighted spaces. Moreover, due to the Hardy term $|x|^{-β}$, some new estimates are established. Based on those works, we give the classification to the linearized problem related to the extremals which has its own interest such as in blow-up analysis. Then we investigate the gradient stability of above inequality by using spectral estimate combined with a compactness argument, which extends the work of Figalli and Zhang (Duke Math. J., 2022) to a weighted case.

math.AP↗

arXiv2022

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

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 to the weighted $p$-Laplace equations. Moreover, we give the classification to the linearized problem related to the radial extremals. As an application, we investigate the gradient type remainder term of related inequality by using spectral estimate combined with a compactness argument which extends the work of Figalli and Zhang (Duke Math. J. 2022) at least for radial case.

math.AP↗

arXiv2022

Some weighted fourth-order Hardy-Henon equations

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-type inequalities: \begin{equation*} \int_{\mathbb{R}^N}|x|^α|Δu|^2 dx \geq S^{rad}(N,α)\left(\int_{\mathbb{R}^N}|x|^{-α}|u|^{p^*_α} dx\right)^{\frac{2}{p^*_α}}, \quad u\in C^\infty_c(\mathbb{R}^N), \end{equation*} where $N\geq 3$, $4-N<α<2$, $p^*_α=\frac{2(N-α)}{N-4+α}$. Then we obtain the explicit form of the unique (up to scaling) radial positive solution $U_{λ,α}$ to the weighted fourth-order Hardy (for $α>0$) or Hénon (for $α<0$) equation: \begin{equation*} Δ(|x|^αΔu)=|x|^{-α} u^{p^*_α-1},\quad u>0 \quad \mbox{in}\quad \mathbb{R}^N. \end{equation*} %Furthermore, we characterize all the solutions to the linearized problem related to above equation at $U_{1,α}$. For $α\neq 0$, it is known the solutions of above equation are invariant for dilations $λ^{\frac{N-4+α}{2}}u(λx)$ but not for translations. However we show that if $α$ is an even integer, there exist new solutions to the linearized problem, which related to above equation at $U_{1,α}$, that "replace" the ones due to the translations invariance. This interesting phenomenon was first shown by Gladiali, Grossi and Neves [Adv. Math. 249, 2013, 1-36] for the second-order Hénon problem. Finally, as applications, we investigate the reminder term of above inequality and also the existence of solutions to some related perturbed equations.

math.AP↗

arXiv2022

Existence and multiplicity of solutions to a Kirchhoff type elliptic system with Trudinger-Moser growth

This paper deals with the existence and multiplicity of solutions for a class of Kirchhoff type elliptic system involving the Trudinger-Moser exponential growth nonlinearities. We first study the existence of solutions for the following system \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -\big(a_1+b_1\|u\|^{2(θ_1-1)}\big)Δu= λH_u(x,u,v)\ \ \ &\ \mbox{in}\ \ \ Ω,\\[2mm] -\big(a_2+b_2\|v\|^{2(θ_2-1)}\big)Δv= λH_v(x,u,v)\ \ \ &\ \mbox{in}\ \ \ Ω,\\[2mm] u=0, v=0\ \ \ \ &\ \mbox{on}\ \ \ \partialΩ, \end{array} \right. \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary,\ $\|u\|=\big(\int_Ω|\nabla u|^2dx\big)^{1/2}$, $H_u$ and $H_v$ behave like $e^{β|s|^2}$ when $|s|\rightarrow \infty$ for some $β>0$, $a_1,\ a_2>0$, $b_1,\ b_2> 0$, $θ_1,\ θ_2> 1$ and $λ$ is a positive parameter. In the later part of the paper, we also discuss a new multiplicity result for the above system with a positive parameter induced by the nonlocal dependence. The Kirchhoff term and the lack of compactness of the associated energy functional due to the Trudinger-Moser embedding have to be overcome via some new techniques.

math.AP↗

arXiv2022

Boundary concentration of peak solutions for fractional Schrödinger-Poisson system

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{array}{ll} \varepsilon^{2s}(-Δ)^{s}u+u+ϕu=u^p,\ \ \ &\ \mbox{in}\ Ω,\\[2mm] (-Δ)^{s}ϕ=u^2,\ \ \ &\ \mbox{in}\ Ω,\\[2mm] u=ϕ=0,\ \ \ \ &\ \mbox{in}\ \mathbb{R}^N\setminus Ω, \end{array} \right. \end{eqnarray*} where $s\in(0,1)$, $N>2s$, $p\in (1,\frac{N+2s}{N-2s})$, $Ω$ is a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary, and $(-Δ)^{s}$ is the fractional Laplacian operator, $\varepsilon$ is a small positive parameter. By using the Lyapunov-Schmidt reduction method, we construct a single peak solution $(u_\varepsilon,ϕ_\varepsilon)$ such that the peak of $u_\varepsilon$ is in the domain but near the boundary. In order to characterize the boundary concentration of solutions, which concentrates at an approximate distance $\varepsilon^{2/3}$ away from the boundary $\partialΩ$ as $\varepsilon$ tends to 0, some new estimates and analytic technique are used.

math.AP↗

arXiv2022

On a nonhomogeneous Kirchhoff type elliptic system with the singular Trudinger-Moser growth

The aim of this paper is to study the multiplicity of solutions for the following Kirchhoff type elliptic systems \begin{eqnarray*} \left\{ \arraycolsep=1.5pt \begin{array}{ll} -m\left(\sum^k_{j=1}\|u_j\|^2\right)Δu_i=\frac{f_i(x,u_1,\ldots,u_k)}{|x|^β}+\varepsilon h_i(x),\ \ & \mbox{in}\ \ Ω, \ \ i=1,\ldots,k ,\\[2mm] u_1=u_2=\cdots=u_k=0,\ \ & \mbox{on}\ \ \partialΩ, \end{array} \right. \end{eqnarray*} where $Ω$ is a bounded domain in $\mathbb{R}^2$ containing the origin with smooth boundary, $β\in [0,2)$, $m$ is a Kirchhoff type function, $\|u_j\|^2=\int_Ω|\nabla u_j|^2dx$, $f_i$ behaves like $e^{βs^2}$ when $|s|\rightarrow \infty$ for some $β>0$, and there is $C^1$ function $F: Ω\times\mathbb{R}^k\to \mathbb{R}$ such that $\left(\frac{\partial F}{\partial u_1},\ldots,\frac{\partial F}{\partial u_k}\right)=\left(f_1,\ldots,f_k\right)$, $h_i\in \left(\big(H^1_0(Ω)\big)^*,\|\cdot\|_*\right)$. We establish sufficient conditions for the multiplicity of solutions of the above system by using variational methods with a suitable singular Trudinger-Moser inequality when $\varepsilon>0$ is small.

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