TY - JOUR
T1 - Regularity of Positive Solutions for an Integral System on Heisenberg Group
AU - Chen , Weiyang
AU - Chen , Xiaoli
JO - Journal of Mathematical Study
VL - 2
SP - 208
EP - 220
PY - 2014
DA - 2014/06
SN - 47
DO - http://doi.org/10.4208/jms.v47n2.14.05
UR - https://global-sci.org/intro/article_detail/jms/9955.html
KW - Ground state solutions, Heisenberg group, nonlinear integral system.
AB - <p style="text-align: justify;">In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, \begin{equation} \left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.\end{equation} <br/>for $x\in \mathbb{H}^n$, where $0&lt;\alpha&lt;Q=2n+2$, $n\geq3$, $\beta\geq0$, $\alpha+\beta&lt;Q$, and $p,q,r &gt; 1$ satisfying $\frac{1}{p+1} $+ $\frac{1}{q+1} + \frac{1}{r+1} =
\frac{Q+α+β}{Q}.$ We show that positive solution triples $(u,v,w)\in L^{p+1}(\mathbb{H}^n)\times L^{q+1}(\mathbb{H}^n)\times L^{r+1}(\mathbb{H}^n)$ are bounded and they converge to zero when $|x|→∞.$</p>