arXiv subjects
Publications and source records attributed to Meiqing Xu.
In this paper, we develop the method of moving planes entirely in the weak formulation for the fractional $p$-Laplacian in the singular range $1 p-1$. Our results complete those of Chen-Li (Adv. Math., 2018: 735-758), where analogous results were obtained for $C^{1,1}$ solutions in the pointwise sense.
In this paper, we study the non-existence of positive solutions for the following conformal $Q$-curvature equation \begin{equation*} (-Δ)^σu = K(x) u^{\frac{n+2σ}{n-2σ}} \quad \text{in } \mathbb{R}^n, \end{equation*} where $σ\in (0, n/2)$ is a real number. When $σ=1$, this equation reduces to the well-known scalar curvature equation arising from the prescribed scalar curvature problem. For general $σ\in (0, n/2)$, it appears in the study of prescribing $Q$-curvature. We establish Liouville theorems under various assumptions on the $Q$-curvature $K(x)$ by developing a unified approach applicable to all $σ\in (0, n/2)$. Our method successfully addresses the challenges posed by the absence of ODE tools in the fractional regime and the lack of a classification of Delaunay-type singular solutions for the general fractional Yamabe equation. Moreover, in the case where $K(x)$ is sign-changing, our result improves upon that of Chen-Li (Comm. Pure Appl. Math. 1995: 657-667) even in the classical setting $σ= 1$, by removing the asymptotic assumption on the solution at infinity.
In this paper, we study isolated singularities of the following semilinear elliptic equation $-Δu+\frac12 x\cdot \nabla u+\frac{1}{q-1}u-u^q=0$ in $Ω\setminus \{0\}$, where $n\ge 3$, $Ω\subset \mathbb{R}^n$ is a domain, $0 \in Ω$ and $q>1$. This equation arises in the study of blow-up profiles of semilinear heat equations. For $\frac{n}{n-2}< q < \frac{n+2}{n-2}$, we establish a complete classification of isolated singularities for nonnegative solutions and characterize the precise asymptotic behavior of singular solutions. Our results improve those of Guedda and Kirane (Trans. Amer. Math. Soc., 1995: 3595-3603), where analogous results were obtained only for radially symmetric positive solutions. In addition, we also derive the asymptotic behavior of solutions in the Serrin critical case $q=\frac{n}{n-2}$ and the supercritical case $q>\frac{n+2}{n-2}$.
This paper works on the direct method of moving spheres and establishes a Liouville-type theorem for the fractional elliptic equation \[ (-Δ)^{α/2} u =f(u) ~~~~~~ \text{in } \mathbb{R}^{n} \] with general non-linearity. One of the key improvement over the previous work is that we do not require the usual Lipschitz condition. In fact, we only assume the structural condition that $f(t) t^{- \frac{n+α}{n-α}}$ is monotonically decreasing. This differs from the usual approach such as Chen-Li-Li (Adv. Math. 2017), which needed the Lipschitz condition on $f$, or Chen-Li-Zhang (J. Funct. Anal. 2017), which relied on both the structural condition and the monotonicity of $f$. We also use the direct moving spheres method to give an alternative proof for the Liouville-type theorem of the fractional Lane-Emden equation in a half space. Similarly, our proof does not depend on the integral representation of solutions compared to existing ones. The methods developed here should also apply to problems involving more general non-local operators, especially if no equivalent integral equations exist.
In this paper, we study the semilinear integro-differential equations \begin{equation*} \mathcal{L}_{K}u(x)\equiv C_n\text{P.V.}\int_{\R^n}\left(u(x)-u(y)\right)K(x-y)dy=f(x,u), \end{equation*} and the full nonlinear integro-differential equations \begin{equation*} F_{G,K}u(x)\equiv C_n\text{P.V.}\int_{\R^n}G(u(x)-u(y))K(x-y)dy=f(x,u), \end{equation*} where $K(\cdot)$ is a symmetric jumping kernel and $K(\cdot)\geq C|\cdot|^{-n-α}$, $G(\cdot)$ is some nonlinear function without non-degenerate condition. We adopt the direct method of moving planes to study the symmetry and monotonicity of solutions for the integro-differential equations, and investigate the limit of some non-local operators $\mathcal{L}_{K}$ as $α\to2.$ Our results extended some results obtained in \cite{CL} and \cite{CLLG}.
In this paper, we focus on the partial differential equation \begin{equation*} (-Δ)^\fracα{2} u(x)=f(x,u(x))\;\;\;\;\text{ in }\mathbb{R}^n, \end{equation*} where $0<α\leq 2$. By the direct method of scaling spheres investigated by Dai and Qin (\cite{dai2023liouville}, \textit{International Mathematics Research Notices, 2023}), we derive a Liouville-type theorem. This mildly extends the previous researches on Liouville-type theorem for the semi-linear equation $ (-Δ)^\fracα{2} u(x)=f(u(x))$ where the nonlinearity $f$ depends solely on the solution $u(x)$, and covers the Liouville-type theorem for Hardy-Hénon equations $(-Δ)^\fracα{2} u(x)=|x|^au^p(x)$.
In this paper, we establish a priori estimates for the positive solutions to a higher-order fractional Laplace equation on a bounded domain by a blowing-up and rescaling argument. To overcome the technical difficulty due to the high-order and fractional order mixed operators, we divide the high-order fractional Laplacian equation into a system, and provide uniform estimates for each equation in the system. Finding a proper scaling parameter for the domain is the crux of rescaling argument to the above system, and the new idea is introduced in the rescaling proof, which may hopefully be applied to many other system problems. In order to derive a contradiction in the blowing-up proof, combining the moving planes method and suitable Kelvin transform, we prove a key Liouville-type theorem under a weaker regularity assumption in a half space.
Let $0<α<2$, $p\geq 1$, $m\in\mathbb{N}_+$. Consider $u$ to be the positive solution of the PDE \begin{equation}\label{abstract PDE} (-Δ)^{\fracα{2}+m} u(x)=u^p(x) \quad\text{in }\mathbb{R}^n. \end{equation} Cao, Dai and Qin( Transactions of the American mathematical society, 2021) showed that, under the condition $u\in\mathcal{L}_α$, the PDE possesses super polyharmonic property $(-Δ)^{k+\fracα{2}}u\geq 0$ for $k=0,1,...,m-1$. In this paper, we show another kind of super polyharmonic property $(-Δ)^k u> 0$ for $k=1,...m$ under different conditions $(-Δ)^mu\in\mathcal{L}_α$ and $(-Δ)^m u\geq 0$. Both kinds of super polyharmonic properties can lead to the equivalence between the PDE and the integral equation $u(x)=\int_{\mathbb{R}^n}\frac{u^p(y)}{|x-y|^{n-2m-α}}dy$.