Regularity and Liouville theorem on an integral equation of Allen-Cahn type

In this paper, we studied an integral equation of the Allen-Cahn type align* u(x) = l +C*∫_Rⁿu(y)(1-|u(y)|²)|1-|u(y)|²|ᵖ⁻² |x-y|n-αdy. align* Here, u:Rⁿ→ Rᵏ, k ≥ 1, n ≥ 2, α ∈ (0,n) and p-1>n/(n-α). In addition, l∈ Rᵏ is a constant vector and C* ≠ 0 is a real constant. This equation is associated with the following fractional-order equation(-Δ)α/2u = u(1-|u|²)|1-|u|²|ᵖ⁻².Both of them play important roles in the study of phase transition in fractal media and the competition model of biological communities. We were concerned about two properties of those equations: One is the Lipschitz regularity of the bounded solutions; the other is the Liouville theorem. The regularity lifting lemma introduced by Ma-Chen-Li and the Pohozaev identity in integral form come into play here.