A Topological Degree Counting for Some Liouville Systems of Mean Field Type

作者：Lin Chang Shou^*; Zhang Lei

来源：Communications on Pure and Applied Mathematics, 2011, 64(4): 556-590.

DOI：10.1002/cpa.20355

Let A = (a(ij))(nxn) be an invertible matrix and A(-1) = (a(ij))(nxn) be the inverse of A. In this paper, we consider the generalized Liouville system
(0.1) Delta(g)u(i) + Sigma(j=1) (n) a(ij) rho(j) (h(j)e(u)j/integral h(j)e(u)j - 1) = 0 in M,
where 0 < h(j) is an element of C-1 (M) and rho(j) is an element of R+, and prove that, under the assumptions of (H-1) and (H-2) (see Introduction), the Leray-Schauder degree of (0.1) is equal to
(-chi(M) + 1) ... (-chi(M) + N)/N!
if rho = (rho 1, ... , rho(n)) satisfies
8 pi N Sigma(i=1) (n) rho i < Sigma (1 <= i, j <= n) aij rho i rho j < 8 pi(N + 1) Sigma(i=1) (n) rho i.
Equation (0.1) is a natural generalization of the classic Liouville equation and is the Euler-Lagrangian equation of the nonlinear function Phi(rho):
Phi(rho)(u) = 1/2 integral(M) Sigma(1 <= i, j <= n) a(ij) del(g) u(j) + Sigma(i=1) (n) integral(M) rho(i) u(i) - Sigma(i=1) (n) rho(i) log integral(M) h(i)e(ui).
The Liouville system (0.1) has arisen in many different research areas in mathematics and physics. Our counting formulas are the first result in degree theory for Liouville systems.

• 出版日期2011-4