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Aliang Xia

Publications and source records attributed to Aliang Xia.

6 recordsLinked to original sources
arXiv2025

Quantitative unique continuation property for fourth-order Baouendi-Grushin type subelliptic operators with a potential

We investigate the quantitative unique continuation property for solutions to $$\Delta^2_{X} u = V u,$$ where $\Delta_{X} = \Delta_{x} + |x|^{2\beta} \Delta_{y}$ ($0 < \beta \leq 1$), with $x \in \mathbb{R}^{m}$ and $y \in \mathbb{R}^{n}$, denotes a class of subelliptic operators of Baouendi-Grushin type. The potential $V$ is assumed to be bounded and satisfy $|Z V| \leq K \psi$ for some constant $K>0$, where $Z= \sum_{i=1}^m x_i \partial_{x_i} + (\beta+1)\sum_{j=1}^n y_j \partial_{y_j}$, $\psi$ is the angle function given by $\psi = \frac{|x|^{2\beta}}{\rho^{2\beta}}$, and $$\rho(x,y) = \left(|x|^{2(\beta+1)} + (\beta+1)^2 |y|^2\right)^{\frac{1}{2(\beta+1)}}$$ defines the associated pseudo-gauge. By adapting Almgren's approach, we establish an almost monotonicity formula for the frequency function. As a consequence, we derive a quantitative unique continuation result for solutions to the fourth-order subelliptic equation.

math.AP↗
arXiv2016

Multiple positive solutions for nonlinear critical fractional elliptic equations involving sign-changing weight functions

In this article, we prove the existence and multiplicity of positive solutions for the following fractional elliptic equation with sign-changing weight functions: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu= a_λ(x)|u|^{q-2}u+b(x)|u|^{2^*_α-1}u &{\rm in}\,\,Ω, u=0\,\,&{\rm in}\,\,\R^N\setminusΩ, \end{array} \right. \end{eqnarray*} where $0<α<1$, $ Ω$ is a bounded domain with smooth boundary in $ \R^N $ with $N>2α$ and $ 2^*_α=2N/(N-2α)$ is the fractional critical Sobolev exponent. Our multiplicity results are based on studying the decomposition of the Nehari manifold and the Ljusternik-Schnirelmann category.

math.AP↗
arXiv2015

Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in R^N involving fractional Laplacian

In this paper, we study the existence and uniqueness of positive solutions for the following nonlinear fractional elliptic equation: \begin{eqnarray*} (-Δ)^αu=λa(x)u-b(x)u^p&{\rm in}\,\,\R^N, \end{eqnarray*} where $ α\in(0,1) $, $ N\ge 2 $, $λ>0$, $a$ and $b$ are positive smooth function in $\R^N$ satisfying \[ a(x)\rightarrow a^\infty>0\quad {\rm and}\quad b(x)\rightarrow b^\infty>0\quad{\rm as}\,\,|x|\rightarrow\infty. \] Our proof is based on a comparison principle and existence, uniqueness and asymptotic behaviors of various boundary blow-up solutions for a class of elliptic equations involving the fractional Laplacian.

math.AP↗
arXiv2015

Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu=f(x,u,v,\nabla u, \nabla v) &{\rm in}\,\,Ω,\\ (-Δ)^αv=g(x,u,v,\nabla u, \nabla v) &{\rm in}\,\,Ω,\\ u=v=0\,\,&{\rm in}\,\,\R^N\setminusΩ, \end{array} \right. \end{eqnarray*} where $(-Δ)^α$ denotes the fractional Laplacian and $ Ω$ is a smooth bounded domain in $ \R^N $. It shown that under some assumptions on $ f $ and $ g $, the problem has at least one positive solution $(u,v)$. Our proof is based on the classical scaling method of Gidas and Spruck and topological degree theory.

math.AP↗
arXiv2015

A Liouville type theorem for Lane-Emden systems involving the fractional Laplacian

We establish a Liouville type theorem for the fractional Lane-Emden system: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-Δ)^αu=v^q&{\rm in}\,\,\R^N,\\ (-Δ)^αv=u^p&{\rm in}\,\,\R^N, \end{array} \right. \end{eqnarray*} where $ α\in(0,1) $, $ N>2α$ and $ p,q $ are positive real numbers and in an appropriate new range. To prove our result we will use the local realization of fractional Laplacian, which can be constructed as Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre \cite{CS}. Our proof is based on a monotonicity argument for suitable transformed functions and the method of moving planes in an infinity half cylinder based on some maximum principles which obtained by some barrier functions and a coupling argument using fractional Sobolev trace inequality.

math.AP↗