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*}
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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.