Zou Wenming | Searcharxiv

arXiv2022

Asymptotic behavior of positive solutions to the Lane-Emden system in dimension two

Consider the Lane-Emden system \begin{equation*}\begin{aligned} &-Δu=v^p,\quad u>0,\quad\text{in}~Ω, &-Δv=u^q,\quad v>0,\quad\text{in}~Ω, &u=v=0,\quad\text{on}~\partialΩ, \end{aligned}\end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb{R}^N$ with $N\geq 2$ and $q\ge p>0$. The asymptotic behavior of {\it least energy solutions} of this system was studied for $N\geq 3$. However, the case $N=2$ is different and remains completely open. In this paper, we study the case $N=2$ with $q=p+θ_p$ and $\sup_pθ_p<+\infty$. Under the following natural condition that holds for least energy solutions $$\limsup_{p\to+\infty} p\int_Ω\nabla u_p\cdot\nabla v_p \mathrm{d} x<+\infty,$$ we give a complete description of the asymptotic behavior of {\it positive solutions} $(u_p,v_p)$ (i.e., not only for least energy solutions) as $p\to+\iy$. This seems the first result for asymptotic behaviors of the Lane-Emden system in the two dimension case.

math.AP↗

arXiv2015

On Elliptic Equations and Systems involving critical Hardy-Sobolev exponents (non-limit case)

Let $Ω\subset \R^N$ ($N\geq 3$) be an open domain (may be unbounded) with $0\in \partialΩ$ and $\partialΩ$ be of $C^2$ at $0$ with the negative mean curvature $H(0)$. By using variational methods, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents, $$\begin{cases} -Δu+λ^*\frac{u}{|x|^{σ_0}}-λ_1 \frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=λ\frac{1}{|x|^{s_2}}|u|^{α-2}u|v|^β\quad &\hbox{in}\;Ω,\\ -Δv+μ^*\frac{v}{|x|^{η_0}}-μ_1 \frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=μ\frac{1}{|x|^{s_2}}|u|^α|v|^{β-2}v\quad &\hbox{in}\;Ω,\\ (u,v)\in D_{0}^{1,2}(Ω)\times D_{0}^{1,2}(Ω), \end{cases}$$ where $ 0\leq σ_0, η_0, s_2<2, s_1\in (0,2);$ the parameters $ λ^*\neq 0, μ^*\neq 0, λ_1>0, μ_1>0, λμ>0$; $α,β>1$ satisfying $α+β\leq 2^*(s_2)$. Here, $2^*(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. We obtain the existence and nonexistence of ground state solution under different specific assumptions. As the by-product, we study \be\lab{zou=a1} \begin{cases} &Δu+λ\frac{u^p}{|x|^{s_1}}+\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}=0\;\quad \hbox{in}\;Ω,\\ &u(x)>0\;\hbox{in}\;Ω,\\ & u(x)=0\;\hbox{on}\;\partialΩ, \end{cases} \ee we also obtain the existence and nonexistence of solution under different hypotheses. In particular, we give a partial answers to a generalized open problem proposed by Y. Y. Li and C. S. Lin (ARMA, 2012). Around the above two types of equation or systems, we systematically study the elliptic equations which have multiple singular terms and are defined on any open domain. We establish some fundamental results. \vskip0.23in {\it Key words:} Elliptic system, Ground state, Hardy-Sobolev exponent.

math.AP↗