Asymptotic behavior of an integral equation of Allen-Cahn type

In this paper, we are concerned with the asymptotic behavior of solutions of an integral equation of the Allen-Cahn type \begin{equation*} u(x)=\overrightarrow {l} +K(x)W_{\beta ,p}[u|u|^{q_1-2}(1-|u|^{q_1})|1-|u|^{q_1}|^{q_2-2}](x) \end{equation*} when $|x| \to \infty$. Here $u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^k$ is uniformly continuous, and $k \geq 1$, $n \geq 2$, $p>1$, $\beta >0$, $p\beta <n$, and $\min \{q_1, q_2\} >2$. In addition, $\overrightarrow {l} \in {\mathbb {R}}^k$ is a constant vector and $K(x)$ is a nonnegative bounded function. We prove that if $u$ or $1-|u|^{q_1}$ has some integrability, then $|\overrightarrow {l}| \in \{0,1\}$ and $u \to \overrightarrow {l}$ when $|x| \to \infty$. Moreover, if $u$ or $1-|u|^{q_1}$ has some better integrability, then $| |u(x)|-|\overrightarrow {l}| |=O(|x|^{(p\beta -n)/(p-1)})$ when $|x| \to \infty$. Here a regularity lifting lemma comes into play in the proof.