Accepted Paper

Inserted: 22 dec 2022
Last Updated: 15 sep 2023

Journal: Advanced Nonlinear Studies
Year: 2023
Doi: http://dx.doi.org/10.1515/ans-2023-0103

Abstract:

This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation} %\tag{$\mathcal P$}\label{MAT1} \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,} % % u,\;v\in\wsp, \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 smooth bounded domains in $\mathbb{R}^N$. For $\Omega=\mathbb{R}^N$ and $\gamma=1$, we show that any ground state solution of the above system has the form $(\lambda U, \tau\lambda V)$ for certain $\tau>0$ and $U,\;V$ are two positive ground state solutions of $(-\Delta_p)^s u =
u
^{p^*_s-2}u$ in $\mathbb{R}^N$. For all $\gamma>0$, we establish existence of a positive radial solution to the above system in balls. For $\Omega=\mathbb{R}^N$, we also establish existence of positive radial solutions to the above system in various ranges of $\gamma$.