Kelvin transform for Grushin operators and critical semilinear equations
Kelvin transform for Grushin operators and critical semilinear equations
复制标题
DOI：
10.1215/s0012-7094-05-13115-5
复制
发表时间：
2006-01
影响因子：
2.5
通讯作者：
R. Monti;Daniele Morbidelli
中科院分区：
文献类型：
--
作者：
R. Monti;Daniele Morbidelli
我们研究 R = R × R 中关键方程 xu+ (α + 1)2|x|2α yu = −u(Q+2)/(Q−2) 的正整数解 u = u(x, y)， (1) 其中 (x, y) ∈ Rm×Rk ，α > 0，且 Q = m+ k(α+1)。在本文的第一部分中，利用方程相对于合适的共形反演的不变性，我们证明了解的“球对称”结果。在第二部分中，我们展示如何使用双曲对称论证来减少问题的维度。给定 (1) 的任何正解 u，在对变量 y 进行适当的缩放和平移之后，函数 v(x) = u(x, 0) 满足方程 divx(p∇xv) − qv = −pv(Q+2)/(Q−2), |x| < 1, (2) 具有混合边界条件。这里，p和q是适当的径向函数。在最后一部分中，我们证明如果m = k = 1，则(2)的解是唯一的，并且当m ≥ 3且k = 1时，问题(2)在x径向函数类中具有唯一解。
We study positive entire solutions u = u(x, y) of the critical equation xu+ (α + 1)2|x|2α yu = −u(Q+2)/(Q−2) in R = R × R, (1) where (x, y) ∈ Rm×Rk , α > 0, and Q = m+ k(α+1). In the first part of the article, exploiting the invariance of the equation with respect to a suitable conformal inversion, we prove a “spherical symmetry” result for solutions. In the second part, we show how to reduce the dimension of the problem using a hyperbolic symmetry argument. Given any positive solution u of (1), after a suitable scaling and a translation in the variable y, the function v(x) = u(x, 0) satisfies the equation divx(p∇xv) − qv = −pv(Q+2)/(Q−2), |x| < 1, (2) with a mixed boundary condition. Here, p and q are appropriate radial functions. In the last part, we prove that if m = k = 1, the solution of (2) is unique and that for m ≥ 3 and k = 1, problem (2) has a unique solution in the class of x-radial functions.