LAUSR

In this paper, we study the asymptotic behavior of solutions of an integral equation of the Allen–Cahn type in Rn u(x)=l⃗+C*∫Rnu(y)(1−|u(y)|2)|1−|u(y)|2|p−2|x−y|n−αdy , when |x| → ∞. Here u:Rn→Rk is uniformly continuous, and k ⩾ 1, n ⩾ 2, α ∈ (0, n) and }\frac{n}{n-\alpha }$?> p−1>nn−α . In addition, l⃗∈Rk is a constant vector and C * is a real constant. If 1−|u|2∈Ls(Rn) for some s ∈ [1, ∞), we know that |u| → 1 when |x| → ∞. Furthermore, we prove that if 1−|u|2∈Ls(Rn) for some s∈[1,nα(p−1)) , then u→l⃗ when |x| → ∞, and hence |l⃗|=1 . When 1−|u|2∈Ls(Rn) for some s∈[1,nα(p−2)) , then there exists some positive constant C such that |1 − |u(x)|2| ⩽ C|x| α−n for large |x|. Here the Harnack type estimate and the regularity lifting lemma come into play in those proofs.