Asymptotic Symmetry and Local Behaviors of Solutions to a Class of Anisotropic Elliptic Equations

作者：Hsia Chun Hsiung^*; Lin Chang Shou; Wang Zhi Qiang

来源：Indiana University Mathematics Journal, 2011, 60(5): 1623-1653.

DOI：10.1512/iumj.2011.60.4376

• Hardy-Sobolev inequality
• Caffarelli-Kohn-Nirenberg inequality
• moving planes method

Let 0 <= a < (N - 2)/2, a <= b < a + 1, p = 2N/(N - 2 + 2(b - a)), and B-1 = B-1(0) be the unit ball in R-N, where N >= 3. We first prove that any positive solution u(x) is an element of C-2(B-1 \ {0}) of the equation
(0.1) {-div(vertical bar x vertical bar(-2a) del u) = vertical bar x vertical bar(-bp)u(p-1) for x is an element of B-1 \ {0},
0 is not a removable singularity of u(x),
is asymptotically symmetric with respect to the origin, i.e.,
u(x) = u(0)(vertical bar x vertical bar)(1 + O(vertical bar x vertical bar(epsilon)) as vertical bar x vertical bar -> 0,
where u(0)(x) = u(0)(vertical bar x vertical bar) is an element of C-2 (R-N \ {0}) is an entire solution of (0.1) and epsilon > 0. Equation (0.1) is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality.
For a = b < 0 and p = 2N / (N - 2), we show there is no positive solution of the equation
(0.2) {-div(vertical bar x vertical bar(-2a)del u) = vertical bar x vertical bar(-bp)u(p-1) in Omega,
u = 0 on partial derivative Omega
in D-a(1,2)(Omega), where Omega subset of R-+(N) is a cone domain satisfying Omega = R+ X omega with omega subset of SN-1 being star-shaped with respect to the north pole on SN-1.

• 出版日期2011