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Nonlinear elliptic equations on the upper half space
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2020-11-30 , DOI: 10.1142/s0219199720500856 Sufang Tang 1 , Lei Wang 2, 3, 4 , Meijun Zhu 4
Communications in Contemporary Mathematics ( IF 1.2 ) Pub Date : 2020-11-30 , DOI: 10.1142/s0219199720500856 Sufang Tang 1 , Lei Wang 2, 3, 4 , Meijun Zhu 4
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In this paper, we shall classify all positive solutions of Δ u = a u p on the upper half space H = ℝ + n with nonlinear boundary condition ∂ u / ∂ t = b u q 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:
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。
更新日期:2020-11-30
中文翻译:
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