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This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional px-operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak solution. This is our first attempt to study this kind of system, in the case of variable-order fractional variable exponents. Our main theorem extends in several directions previous results.

In this paper, we are concerned with the following fractional p-Kirchhoff system with sign-changing nonlinearities: M(∫R2nux-uyp/x-yn+psdxdy)-Δpsu=λa(x)uq-2u+α/(α+β)f(x)uα-2uvβ, in Ω, M(∫R2n|v(x)-v(y)|p/|x-y|n+psdxdy)-Δpsv=μb(x)vq-2v+(β/α+β)f(x)uαvβ-2v, in Ω, and u=v=0, in Rn∖Ω, where Ω is a smooth bounded domain in Rn, n>ps, s∈(0,1), λ, μ are two real parameters, 10, l>0, h≥1,a(x),b(x)∈L(α+β)/(α+β-q)(Ω) are sign changing and either a±=max⁡{±a,0}≢0 or b±=max⁡{±b,0}≢0, f∈L(Ω¯) with f∞=1, and f≥0. Using Nehari manifold method, we prove that the system has at least two solutions with respect to the pair of parameters (λ,μ).