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

Abstract

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} (-Δ)^{s_1}_{p_1} u &=u|u|^{α-1}|v|^{β+1}+f_1(x) \,\mbox{ in }\, Ω, \\ (-Δ)^{s_2}_{p_2} v &=|u|^{α+1}v|v|^{β-1}+f_2(x) \,\mbox{ in }\, Ω, \\ u=v &= 0 \,\mbox{ in }\, \mathbb{R}^N \setminus Ω, \\ \end{aligned} \right. \end{equation*} where $\Omega \subset {\mathbb{R}}^N$ is smooth bounded domain, $s_1,s_2\in (0,1)$, $1<p_1,p_2<\infty$, $N>\max \{p_1{s}_1,p_2{s}_2\}$, $\alpha >-1$ 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 $f_1,f_2$.