Regularity and Liouville theorem on an integral equation of Allen-Cahn type (Q6598504)

In this paper, the authors study an integral equation of Allen-Cahn type:\N\[\Nu(x)=\vec{l}+C_{*}\int_{R^{n}}\frac{u(y)(1-\lvert u(y)\rvert^{2})\lvert 1-\lvert u(y)\rvert^{2}\rvert ^{p-2}}{\lvert x-y\rvert^{n-\alpha}}dy,\tag{1}\N\]\Nwhere \( u:\mathbb R^{n}\rightarrow \mathbb R^{k}, k\geq 1, n\geq 2, \alpha\in (0,n) \) and \( p-1 > n/ (n-\alpha), \vec{l}\in \mathbb R^{k} \) is a constant vector and \( C_{*}\neq 0 \) is a real constant. This equation is associated with a fractional-order equation of the form\N\[\N(-\Delta)^{\alpha/2} u=u(1-\lvert u\rvert^{2})\lvert 1-\lvert u\rvert^{2}\rvert ^{p-2}.\N\]\N\NIn this paper, they have further studied the regularity of solutions and the Liouville theorem:\N\NRegularity Theorem. Let\N\[\Nk\geq 1,n\geq 2, 0< \alpha < n, p-1>n/(n-\alpha) , p\geq 3.\tag{2}\N\]\NSuppose that \( u:\mathbb R^{n}\rightarrow \mathbb R^{k} \) is a solution of (1) with (2). If\N\[\N1-\lvert u\rvert ^{2}\in L^{s}(\mathbb R^{n})\tag{3}\N\]\Nfor some \( s\in [1,n(p-1)/\alpha] \) and \( u\in L^{\infty}(\mathbb R^{n}), \) then \( u \) must be Lipschitz continuous. Moreover, if \( 1<\alpha<n, u \) must be differentiable. Further, if (3) holds for some \( s\in [1,n(p-2)/\alpha] \), then \( \triangledown u \) is also Lipschitz continuous.\N\NLiouville Theorem. Let \( n \in\{3,4,5\}, \) \(\max\{1,n/3\}<\alpha< \min\{2,n/2\} \) and \( p-1 < (n+\alpha)/(n-\alpha). \) Assume that \( u \) is a uniformly continuous solution of (1) and (2) and \( u \) satisfies (3) for some \( s\in[1,n(p-2)/\alpha], \) then \( u\equiv \vec{l} \) with \( \lvert \vec{l} \rvert =1 \).