| Abstract: |
| This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations
\begin{equation*}
\begin{cases}
(-\Delta_p)^s u =|u|^{p^*_s-2}u+ \frac{\gamma\alpha}{p_s^*}|u|^{\alpha-2}u|v|^{\beta}\;\;\text{in}\;\Omega,\
(-\Delta_p)^s v =|v|^{p^*_s-2}v+ \frac{\gamma\beta}{p_s^*}|v|^{\beta-2}v|u|^{\alpha}\;\;\text{in}\;\Omega,
\end{cases}
\end{equation*}
where $s\in(0,1)$, $p\in(1,\infty)$ with $N>sp$, $\alpha,\,\beta>1$ such that $\alpha+\beta = p^*_s:=\frac{Np}{N-sp}$ and $\Omega=\mathbb{R}^N$ or any smooth bounded domains in $\mathbb{R}^N$. When $\Omega=\mathbb{R}^N$ and $\gamma=1$, we show that any ground state solution of the above system has the form $(\lambda U, \tau\lambda V)$ for certain $\tau$ is a positive constant and $U,\;V$ are two positive ground state solutions of $(-\Delta_p)^s u =|u|^{p^*_s-2}u$ in $\mathbb{R}^N$. When $\Omega=\mathbb{R}^N$, we also establish existence of positive radial solutions to the above system in various ranges of $\gamma$. On the other hand, when $\Omega$ is any ball, we show existence of a positive radial solutions to the above system for all $\gamma>0$. |
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