Browsing by Subject "nonlinear system"

Now showing 1 - 2 of 2

Access status: Open Access ,
Existence of critical elliptic systems with boundary singularities
(2013) Yang, Jianfu; Zhou, Yimin
In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponent $\begin{equation*}\label{eq:1}(*) \left\{ \begin{array}{lll} -\Delta u= \frac{2\alpha}{\alpha+\beta}\frac{u^{\alpha-1}v^\beta}{|x|^s}-\lambda u^p, & \quad {\rm in}\quad \Omega,\\[2mm] -\Delta v= \frac{2\beta}{\alpha+\beta}\frac{u^\alpha v^{\beta-1}}{|x|^s}-\lambda v^p, & \quad {\rm in}\quad \Omega,\\[2mm] u\gt 0, v\gt 0, &\quad {\rm in}\quad \Omega,\\[2mm] u=v=0, &\quad {\rm on}\quad \partial\Omega, \end{array} \right. \end{equation*}$ where $N\geq 4$ and $\Omega$ is a $C^1$ bounded domain in $\mathbb{R}^N$ with $0\in\partial\Omega$. $0\lt s \lt 2$, $\alpha+\beta=2^*(s)=\frac{2(N-s)}{N-2}$, $\alpha,\beta\gt 1$, $\lambda\gt 0$ and $1 \lt p\lt \frac{N+2}{N-2}$. The case when 0 belongs to the boundary of $\Omega$ is closely related to the mean curvature at the origin on the boundary. We show in this paper that problem $(*)$ possesses at least a positive solution.
Access status: Open Access ,
The existence of consensus of a leader-following problem with Caputo fractional derivative
(Wydawnictwa AGH, 2019) Schmeidel, Ewa
In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained.