arXiv subjects

Liang-Gen Hu

Publications and source records attributed to Liang-Gen Hu.

3 recordsLinked to original sources

arXiv2015

Liouville type theorems for stable solutions of the weighted elliptic system

We examine the weighted elliptic system \begin{equation*} \begin{cases} -Δu=(1+|x|^2)^{\fracα{2}} v,\\ -Δv=(1+|x|^2)^{\fracα{2}} u^p, \end{cases} \quad \mbox{in}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$, $p>1$ and $α>0$. We prove Liouville type results for the classical positive (nonnegative) stable solutions in dimension $N<\ell+\dfrac{α(\ell-2)}{2}$ ($N <\ell+\dfrac{α(\ell-2)(p+3)}{4(p+1)}$) and $\ell \ge 5$, $p \in (1,p_*(\ell))$. In particular, for any $p>1$ and $α> 0$, we obtain the nonexistence of classical positive (nonnegative) stable solutions for any $N \le 12+5 α$ ($N\le 12+\dfrac{5α(p+3)}{2(p+1)}$).

math.AP↗

arXiv2014

Monotonicity formula and Liouville-type theorems of stable solution for the weighted elliptic system

In this paper, we are concerned with the weighted elliptic system \begin{equation*} \begin{cases} -Δu=|x|^β v^{\vartheta},\\ -Δv=|x|^α |u|^{p-1}u, \end{cases}\quad \mbox{in}\;\ Ω, \end{equation*}where $Ω$ is a subset of $\mathbb{R}^N$, $N \ge 5$, $α>-4$, $0 \le β\le \dfrac{N-4}{2}$, $p>1$ and $\vartheta=1$. We first apply Pohozaev identity to construct a monotonicity formula and find their certain equivalence relation. By the use of {\it Pohozaev identity}, {\it monotonicity formula} of solutions together with a {\it blowing down} sequence, we prove Liouville-type theorems of stable solutions for the weighted elliptic system (whether positive or sign-changing) in the higher dimension.

math.AP↗

arXiv2013

Liouville-type theorems for the fourth order nonlinear elliptic equation

In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equation {equation*} Δ^2 u=|x|^a |u|^{p-1}u\;\ {in}\;\ Ω, {equation*}where $a \ge 0$, $p>1$ and $Ω\subset \mathbb{R}^n$ is an unbounded domain of $\mathbb{R}^n$, $n \ge 5$. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the {\it Pohozaev-type identity}, {\it monotonicity formula} of solutions and a {\it blowing down} sequence, which is used to obtain sharper results.

math.AP↗