Asymptotic estimates for an integral equation in theory of phase transition

In this paper, we study the asymptotic behavior of solutions of an integral equation of the Allen–Cahn type in $R^n$ $u\left(x\right)=\overrightarrow{l}+C_{\ast }{\int }_{R^n}\frac{u\left(y\right)(1-|u\left(y\right){|}^2)|1-|u\left(y\right){|}^2{|}^{p-2}}{|x-y{|}^{n-\alpha }}dy$, when |x| → ∞. Here $u:R^n\to R^k$ is uniformly continuous, and k ⩾ 1, n ⩾ 2, α ∈ (0, n) and $p-1>\frac{n}{n-\alpha }$. In addition, $\overrightarrow{l}\in R^k$ is a constant vector and C _* is a real constant. If $1-|u{|}^2\in L^s\left(R^n\right)$ for some s ∈ [1, ∞), we know that |u| → 1 when |x| → ∞. Furthermore, we prove that if $1-|u{|}^2\in L^s\left(R^n\right)$ for some $s\in [1,\frac{n}{\alpha }(p-1))$, then $u\to \overrightarrow{l}$ when |x| → ∞, and hence $\left|\overrightarrow{l}\right|=1$. When $1-|u{|}^2\in L^s\left(R^n\right)$ for some $s\in [1,\frac{n}{\alpha }(p-2))$, 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. (paper)