Malaysian Journal of Mathematical Sciences, September 2021, Vol. 15, No. 3
Existence and Uniqueness of Solutions for a Nonlinear Fractional Elliptic System
Dob, S., Lakhal, H., and Maouni, M.
Corresponding Email: dobsara@yahoo.com
Received date: 21 September 2020
Accepted date: 26 April 2021
Abstract:
In this article, we study the existence and uniqueness of weak solution for the non-linear fractional elliptic system
\begin{equation*}
\begin{cases}
(-\Delta)^s \varphi(z)=p(z,\varphi(z),\phi(z))\quad \text{ in} \quad \Omega,\\
(-\Delta)^s \phi(z)=k(z,\varphi(z),\phi(z))\quad \text{ in} \quad \Omega,\\
\varphi=\phi=0\quad\quad\quad\quad\quad\quad\quad \quad
\text{ on}\quad \mathbb{R}^n \setminus\Omega,
\end{cases}
\end{equation*}
with $s \in (0, 1)$ and $\Omega$ is an open bounded subset of $\mathbb{R}^n$. We use the Schauder fixed point theorem to prove the existence of solution under suitable assumptions on the nonlinearities $p$ and $k$, and the contraction principle to prove the existence and uniqueness of solution in a particular case.
Keywords: nonlinear elliptic equations; fractional Laplacian; weak solution
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