A Liouville-type theorem for an integral equation on a half-space \(\mathbb R_+^n\)
The following integral equation on the \(n\)-dimensional upper half Euclidean space \(\mathbb{R}_{+}^n\) is studied \[ u(x)=\int\limits_{\mathbb{R}_{+}^n}\left(\frac{1}{|x-y|^{n-\alpha}}-\frac{1}{|x^*-y|^{n-\alpha}}\right)u^p(y)dy, \quad u(x)>0,\quad x\in \mathbb{R}_{+}^n, \] where \(\alpha\) is an even number satisfying \(0<\alpha<n\); \(x^*=(x_1,x_2,\dots,x_{n-1},-x_n)\) is the reflection of the point \(x\) about the \(\partial \mathbb{R}_{+}^n\). The authors use the moving planes method in integral forms introduced by Chen-Li-Ou to establish a Liouville-type theorem for this equation, which is closely related to the higher-order differential equation with Navier boundary conditions: \[ (-\Delta)^{\alpha/2}u=u^p, \text{ in } \mathbb{R}_{+}^n; \] \[ u=(-\Delta)u=\cdots=(-\Delta)^{\alpha/2-1}, \text{ on } \partial \mathbb{R}_{+}^n. \]
scientific article; zbMATH DE number 6015291
scientific article