A system of integral equations on half space
In the half-space \({\mathbb R}_+^n=\left\{x\in{\mathbb R}^n:x_n>0\right\}\) the following system of nonlinear integral equations is studied: \[ u(x)=\int_{{\mathbb R}_+^n}\left({1\over\| x-y\|^{n-\alpha}}- {1\over\| x^*-y\|^{n-\alpha}}\right)v(y)^qdy, \] \[ v(x)=\int_{{\mathbb R}_+^n}\left({1\over\| x-y\|^{n-\alpha}}- {1\over\| x^*-y\|^{n-\alpha}}\right)u(y)^pdy. \] Here, \(x^*\) denotes the reflection of \(x\) about the plane \(x_n=0\), \(p,q>1\), and \[ {1\over q+1}+{1\over p+1}={n-\alpha\over n}. \] It is shown that any pair of positive solutions \(u\in L^{p+1}({\mathbb R}_+^n)\), \(v\in L^{q+1}({\mathbb R}_+^n)\) is bounded and continuous. Moreover, using the method of moving planes, it is shown that \((u,v)\) is rotationally symmetric about some line parallel to the \(x_n\)-axis, and that any smooth solution satisfies on \({\mathbb R}_+^n\) the system \[ (-\Delta)^{\alpha/2}u(x)=v(x)^q \] \[ (-\Delta)^{\alpha/2}v(x)=u(x)^p \] and on \(\partial{\mathbb R}_+^n\) the boundary conditions \[ (-\Delta)^ku(x)=(-\Delta)^kv(x)=0 \] for \(k=0,\ldots,{\alpha\over2}-1\).
scientific article; zbMATH DE number 5904478
scientific article