Year: 2014
Author: Weiyang Chen, Xiaoli Chen
Journal of Mathematical Study, Vol. 47 (2014), Iss. 2 : pp. 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 $\mathbb{H}^n$, \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}
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 $|x|→∞.$
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jms.v47n2.14.05
Journal of Mathematical Study, Vol. 47 (2014), Iss. 2 : pp. 208–220
Published online: 2014-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Ground state solutions Heisenberg group nonlinear integral system.
Author Details
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Existence of Positive Solutions to Nonlinear Integral Equations with Weights on the Bounded Domains of the Heisenberg Group inSubcritical Case
陈, 佳妮
Advances in Applied Mathematics, Vol. 11 (2022), Iss. 04 P.1764
https://doi.org/10.12677/AAM.2022.114193 [Citations: 0]