Volume 28, Issue 1
Classification of Solutions to a Critically Nonlinear System of Elliptic Equations on Euclidean Half-Space

J. Part. Diff. Eq., 28 (2015), pp. 74-94.

Published online: 2015-03

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• Abstract

For $N\geq 3$ and non-negative real numbers $a_{ij}$ and $b_{ij}$ ($i,j= 1, \cdots, m$), the semi-linear elliptic system\begin{equation*} \begin{cases}\Delta u_i+\prod\limits_{j=1}^m u_j^{a_{ij}}=0,\text{in}\mathbb{R}_+^N,\\ \dfrac{\partial u_i}{\partial y_N}=c_i\prod\limits_{j=1}^m u_j^{b_{ij}},\text{on} \partial\mathbb{R}_+^N,\end{cases}\qquad i=1,\cdots,m,\end{equation*} % is considered, where $\mathbb{R}_+^N$ is the upper half of $N$-dimensional Euclidean space. Under suitable assumptions on the exponents $a_{ij}$ and $b_{ij}$, a classification theorem for the positive $C^2(\mathbb{R}_+^N)\cap C^1(\overline{R_+^N})$-solutions of this system is proven.

• Keywords

Nonlinear elliptic systems

• AMS Subject Headings

35J57, 35J66, 35K57

@Article{JPDE-28-74, author = {Gluck , Mathew R. and Zhang , Lei}, title = {Classification of Solutions to a Critically Nonlinear System of Elliptic Equations on Euclidean Half-Space}, journal = {Journal of Partial Differential Equations}, year = {2015}, volume = {28}, number = {1}, pages = {74--94}, abstract = {For $N\geq 3$ and non-negative real numbers $a_{ij}$ and $b_{ij}$ ($i,j= 1, \cdots, m$), the semi-linear elliptic system\begin{equation*} \begin{cases}\Delta u_i+\prod\limits_{j=1}^m u_j^{a_{ij}}=0,\text{in}\mathbb{R}_+^N,\\ \dfrac{\partial u_i}{\partial y_N}=c_i\prod\limits_{j=1}^m u_j^{b_{ij}},\text{on} \partial\mathbb{R}_+^N,\end{cases}\qquad i=1,\cdots,m,\end{equation*} % is considered, where $\mathbb{R}_+^N$ is the upper half of $N$-dimensional Euclidean space. Under suitable assumptions on the exponents $a_{ij}$ and $b_{ij}$, a classification theorem for the positive $C^2(\mathbb{R}_+^N)\cap C^1(\overline{R_+^N})$-solutions of this system is proven. }, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v28.n1.7}, url = {http://global-sci.org/intro/article_detail/jpde/5103.html} }
TY - JOUR T1 - Classification of Solutions to a Critically Nonlinear System of Elliptic Equations on Euclidean Half-Space AU - Gluck , Mathew R. AU - Zhang , Lei JO - Journal of Partial Differential Equations VL - 1 SP - 74 EP - 94 PY - 2015 DA - 2015/03 SN - 28 DO - http://doi.org/10.4208/jpde.v28.n1.7 UR - https://global-sci.org/intro/article_detail/jpde/5103.html KW - Nonlinear elliptic systems AB - For $N\geq 3$ and non-negative real numbers $a_{ij}$ and $b_{ij}$ ($i,j= 1, \cdots, m$), the semi-linear elliptic system\begin{equation*} \begin{cases}\Delta u_i+\prod\limits_{j=1}^m u_j^{a_{ij}}=0,\text{in}\mathbb{R}_+^N,\\ \dfrac{\partial u_i}{\partial y_N}=c_i\prod\limits_{j=1}^m u_j^{b_{ij}},\text{on} \partial\mathbb{R}_+^N,\end{cases}\qquad i=1,\cdots,m,\end{equation*} % is considered, where $\mathbb{R}_+^N$ is the upper half of $N$-dimensional Euclidean space. Under suitable assumptions on the exponents $a_{ij}$ and $b_{ij}$, a classification theorem for the positive $C^2(\mathbb{R}_+^N)\cap C^1(\overline{R_+^N})$-solutions of this system is proven.