Maximum and comparison principles for degenerate elliptic systems and some applications

In this paper we develop a detailed study on maximum and comparison principles related to the following nonlinear eigenvalue problem {-Delta(p)u = lambda a vertical bar v vertical bar(beta 1-1)v in Omega; -Delta(q)v = mu b (x)vertical bar u vertical bar(beta 2-1)u in Omega; u=v=0 on partial derivative Omega, where p, q is an element of (1, infinity), beta(1), beta(2 )> 0 satisfy beta(1)beta(2) = (p - 1)(q - 1), Omega subset of R-n is a bounded domain with C-2-boundary, a, b is an element of L-infinity (Omega) are given functions, both assumed to be strictly positive on compact subsets of Q, and Delta(p) and Delta(q) are quasilinear elliptic operators, stand for p-Laplacian and q-Laplacian, respectively. We classify all couples (lambda,mu) is an element of R(2)such that both the (weak and strong) maximum and comparison principles corresponding to the above system hold in Omega. Explicit lower bounds for principal eigenvalues of this system in terms of the measure of Omega are also proved. As application, given lambda, mu >= 0 we measure explicitly how small has to be vertical bar Omega vertical bar so that weak and strong maximum principles associated to the above problem hold in Omega. (C) 2020 Elsevier Inc. All rights reserved.