Maximum Principles and ABP Estimates to Nonlocal Lane-Emden Systems and Some Consequences

This paper deals with maximum principles depending on the domain and ABP estimates associated to the following Lane-Emden system involving fractional Laplace operators: {(-Delta)(s)u = lambda rho(x)vertical bar v vertical bar(alpha-1)v in Omega, (-Delta)(t)u = lambda tau(x)vertical bar u vertical bar(beta-1)v in Omega, u = v = 0 in R-n\Omega, where s, t is an element of (0, 1), alpha, beta > 0 satisfy alpha beta = 1, Omega Zeta is a smooth bounded domain in R-n, n >= 1, and rho and tau are continuous functions on (Omega) over bar and positive in Omega. We establish some maximum principles depending on Omega. In particular, we explicitly characterize the measure of Omega for which the maximum principles corresponding to this problem hold in Omega. For this, we derived an explicit lower estimate of principal eigenvalues in terms of the measure of Omega. Aleksandrov-Bakelman-Pucci (ABP) type estimates for the above systems are also proved. We also show the existence of a viscosity solution for a nonlinear perturbation of the nonhomogeneous counterpart of the above problem with polynomial and exponential growths. As an application of the maximum principles, we measure explicitly how small vertical bar Omega vertical bar has to be to ensure the positivity of the obtained solutions.