In this paper, we investigate the fractional p-Kirchhoff -type system:
{M(∫R2N|u(x)−u(y)|p|x−y|N+psdxdy)(−Δ)psu=μg(x)|u|β−2u+aa+bh(x)|u|a−2u|v|b,in Ω,M(∫R2N|v(x)−v(y)|p|x−y|N+psdxdy)(−Δ)psv=σf(x)|v|β−2v+ba+bh(x)|v|b−2v|u|a,in Ω,u=v=0,in RN∖Ω,\documentclass[12pt]{minimal}
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\begin{document} $$\begin{aligned} \textstyle\begin{cases} M (\int_{{ \mathbb {R} }^{2N}}\frac{\vert u(x)-u(y) \vert ^{p}}{\vert x-y \vert ^{N+ps}}\,dx\,dy )(- \Delta )^{s}_{p}u=\mu g(x)\vert u \vert ^{\beta -2}u+\frac{a}{a+b}h(x)\vert u \vert ^{a-2}u\vert v \vert ^{b},&\mbox{in } \Omega , \\ M (\int_{{ \mathbb {R} }^{2N}}\frac{\vert v(x)-v(y) \vert ^{p}}{\vert x-y \vert ^{N+ps}}\,dx\,dy )(- \Delta )^{s}_{p}v=\sigma f(x)\vert v \vert ^{\beta -2}v+\frac{b}{a+b}h(x)\vert v \vert ^{b-2}v\vert u \vert ^{a},&\mbox{in } \Omega , \\ u=v=0,&\mbox{in } { \mathbb {R} }^{N}\setminus \Omega , \end{cases}\displaystyle \end{aligned}$$ \end{document} where Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$\Omega \subset \mathbb{R}^{N}$\end{document} is a smooth bounded domain, (−Δ)ps\documentclass[12pt]{minimal}
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\begin{document}$(-\Delta )^{s}_{p}$\end{document} is the fractional p-Laplacian operator with 0<s<1<p\documentclass[12pt]{minimal}
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\begin{document}$0< s<1<p$\end{document} and ps<N\documentclass[12pt]{minimal}
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\begin{document}$ps< N $\end{document}. a>1\documentclass[12pt]{minimal}
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\begin{document}$a>1$\end{document}, b>1\documentclass[12pt]{minimal}
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\begin{document}$b>1$\end{document} satisfy 2<a+b<ps∗\documentclass[12pt]{minimal}
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\begin{document}$2< a+b< p_{s}^{*}$\end{document}. 1<β<ps∗\documentclass[12pt]{minimal}
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\begin{document}$1<\beta <p_{s}^{*}$\end{document}, ps∗=NpN−ps\documentclass[12pt]{minimal}
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\begin{document}$p_{s}^{*}=\frac{Np}{N-ps}$\end{document} is the fractional critical exponent. μ, σ are two real parameters. M(t)=k+λtτ\documentclass[12pt]{minimal}
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\begin{document}$M(t)=k+\lambda t^{\tau }$\end{document}, k>0\documentclass[12pt]{minimal}
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\begin{document}$k>0$\end{document}, λ, τ≥0\documentclass[12pt]{minimal}
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\begin{document}$\tau \geq 0$\end{document}, τ=0\documentclass[12pt]{minimal}
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\begin{document}$\tau =0$\end{document} if and only if λ=0\documentclass[12pt]{minimal}
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\begin{document}$\lambda =0$\end{document}. The weight functions g, f, h change sign in Ω and satisfy suitable conditions. By using the Nehari manifold method, it is proved that the system has at least two solutions provided that 2<a+b<p≤p(τ+1)<β<ps∗\documentclass[12pt]{minimal}
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\begin{document}$2< a+b< p\leq p(\tau +1)<\beta <p_{s}^{*}$\end{document} and (μ,σ)\documentclass[12pt]{minimal}
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\begin{document}$(\mu ,\sigma )$\end{document} belongs to a certain subset of R2\documentclass[12pt]{minimal}
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\begin{document}$\mathbb {R} ^{2}$\end{document}. Also, by using the mountain pass theorem, we prove that there exist λ1≥λ0\documentclass[12pt]{minimal}
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\begin{document}$\lambda _{1}\geq \lambda_{0}$\end{document} such that the system admits at least a nontrivial solution for λ∈(0,λ0)\documentclass[12pt]{minimal}
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\begin{document}$\lambda \in (0,\lambda_{0})$\end{document} and no nontrivial solution for λ>λ1\documentclass[12pt]{minimal}
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\begin{document}$\lambda >\lambda_{1}$\end{document} under the assumptions μ=σ=0\documentclass[12pt]{minimal}
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\begin{document}$\mu =\sigma =0$\end{document} and p<a+b<min{p(τ+1),ps∗}\documentclass[12pt]{minimal}
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\begin{document}$p< a+b<\min \{p(\tau +1),p_{s}^{*}\}$\end{document}.
