In this paper, we shall classify all positive solutions of Δu = aup on the upper half space H = ℝ+n with nonlinear boundary condition ∂u/∂t = buq on ∂H for parameters a > 0 and b < 0. We will prove that for p ≥ (n + 2)/(n − 2), 1 ≤ q < n/(n − 2) or p > (n + 2)/(n − 2), 1 ≤ q ≤ n/(n − 2) (and n ≥ 3) all positive solutions are functions of last variable; for p = (n + 2)/(n − 2),q = n/(n − 2) (and n ≥ 3) positive solutions must be either some functions depending only on last variable, or radially symmetric functions.

中文翻译：

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上半空间的非线性椭圆方程

在本文中，我们将对所有正解进行分类Δ你 = 一种你p在上半空间H = ℝ+n具有非线性边界条件∂你/∂吨 = b你q在∂H对于参数一种 > 0和b < 0. 我们将证明对于p ≥ (n + 2)/(n - 2), 1 ≤ q < n/(n - 2)要么p > (n + 2)/(n - 2),1 ≤ q ≤ n/(n - 2)（和n ≥ 3) 所有正解都是最后一个变量的函数；为了p = (n + 2)/(n - 2),q = n/(n - 2)（和n ≥ 3) 正解必须是仅取决于最后一个变量的某些函数，或者是径向对称函数。