Volume 47, Issue 2
Regularity of Positive Solutions for an Integral System on Heisenberg Group

J. Math. Study, 47 (2014), pp. 208-220.

Published online: 2014-06

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• Abstract

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, $$\left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.$$
for $x\in \mathbb{H}^n$, where $0<\alpha<Q=2n+2$, $n\geq3$, $\beta\geq0$, $\alpha+\beta<Q$, and $p,q,r > 1$ 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 $|x|→∞.$

• Keywords

Ground state solutions, Heisenberg group, nonlinear integral system.

45E10, 45G05

xiaowei19901207@126.com (Weiyang Chen)

littleli_chen@163.com (Xiaoli Chen)

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@Article{JMS-47-208, author = {Weiyang and Chen and xiaowei19901207@126.com and 13257 and Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China and Weiyang Chen and Xiaoli and Chen and littleli_chen@163.com and 13258 and Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China and Xiaoli Chen}, title = {Regularity of Positive Solutions for an Integral System on Heisenberg Group}, journal = {Journal of Mathematical Study}, year = {2014}, volume = {47}, number = {2}, pages = {208--220}, abstract = {

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, $$\left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.$$
for $x\in \mathbb{H}^n$, where $0<\alpha<Q=2n+2$, $n\geq3$, $\beta\geq0$, $\alpha+\beta<Q$, and $p,q,r > 1$ 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 $|x|→∞.$

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v47n2.14.05}, url = {http://global-sci.org/intro/article_detail/jms/9955.html} }
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 -

In this paper, we are concerned with the properties of positive solutions of the following nonlinear integral systems on the Heisenberg group $\mathbb{H}^n$, $$\left\{\begin{array}{ll} u(x)=\int_{\mathbb{H}^n}\frac{v^{q}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ v(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)w^{r}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ w(x)=\int_{\mathbb{H}^n}\frac{u^{p}(y)v^{q}(y)}{|x^{-1}y|^\alpha|y|^\beta}\,dy,\\ \end{array}\right.$$
for $x\in \mathbb{H}^n$, where $0<\alpha<Q=2n+2$, $n\geq3$, $\beta\geq0$, $\alpha+\beta<Q$, and $p,q,r > 1$ 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 $|x|→∞.$

Weiyang Chen & Xiaoli Chen. (2019). Regularity of Positive Solutions for an Integral System on Heisenberg Group. Journal of Mathematical Study. 47 (2). 208-220. doi:10.4208/jms.v47n2.14.05
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