Remarks on an Integral Equation of Landau–Lifschitz Type

In this paper, we are concerned with the asymptotic behavior and the Liouville theorem for the nonlinear integral equation

where \(u \in C^{\infty}(\mathbb{R}^{n},\mathbb{S}^{k-1})\) with \(n \geq 3\), \(k \geq 2\), \(\alpha \in (0,n/2)\), \(\mathbb{S}^{k-1}=\{u=(u_1,u_2,\cdots,u_k) \in \mathbb{R}^{k}; |u|=1\}\), \(e_k=(0,\cdots,0,1)\), \(C_*\) is a positive constant, and \(\ell \in \mathbb{R}^{k}\) is a constant vector. We prove that, if \(u_k \in L^2(\mathbb{R}^n)\) and \(\nabla u \in L^2(\mathbb{R}^n) \cap L^\infty(\mathbb{R}^n)\), then \(u \to \ell\) as \(|x| \to \infty\) with \(\ell_k=0\). Moreover, if \(\alpha \in (1,n/2)\), then \(u \equiv \ell\) on \(\mathbb{R}^n\).

DOI 10.1134/S1061920825600576