AN INTEGRAL SYSTEM AND THE LANE-EMDEN CONJECTURE

作者：Chen Wenxiong^*; Li Congming

来源：Discrete and Continuous Dynamical Systems, 2009, 24(4): 1167-1184.

DOI：10.3934/dcds.2009.24.1167

• Lane-Emden conjecture
• elliptic PDE systems
• integral systems
• symmetry
• non-existence
• moving planes

We consider the system of integral equations in R(n):
{v(x) = integral R(n) vertical bar(x-y vertical bar n-mu)/(1) u(q)(y)dy v(x) = integral Rn vertical bar(x-y vertical bar n-mu)/(1) u(p)(y)dy
with 0 < mu n. Under some integrability conditions, we obtain radial symmetry of positive solutions by using the method of moving planes in integral forms. In the special case when mu = 2, we show that the integral system is equivalent to the elliptic PDE system
{-Delta u = u(q)(x) -Delta v = u(p)(x)
in R(n). Our symmetry result, together with non-existence of radial solutions by Mitidieri [30], implies that, under our integrability conditions, the PDE system possesses no positive solution in the subcritical case. This partially solved the well-known Lane-Emden conjecture.

• 出版日期2009-8
• 单位河南师范大学