Non-homogeneous $(p_1,p_2)$-fractional Laplacian systems with lack of compactness

The present paper studies the existence of weak solutions for the following type of non-homogeneous system of equations \begin{equation*} (S) \left{\begin{aligned} (-\Delta)^{s_1}{p_1} u &=u|u|^{\alpha-1}|v|^{\beta+1}+f_1(x) ,\mbox{ in }, \Omega, \ (-\Delta)^{s_2}{p_2} v &=|u|^{\alpha+1}v|v|^{\beta-1}+f_2(x) ,\mbox{ in }, \Omega, \ u=v &= 0 ,\mbox{ in }, $\mathbb{R}$^N \setminus \Omega, \ \end{aligned} \right. \end{equation*} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is smooth bounded domain, $s_1,s_2\in (0,1)$, $1<p_1,p_2<\mathrm{\infty }$, $N>\max \{p_1{s}_1,p_2{s}_2\}$, $\alpha >-1$ and $\beta >-1$. We employ the variational techniques where the associated energy functional is minimized over Nehari manifold set while imposing appropriate bound on dual norms of $f_1,f_2$.