Maximum and comparison principles for degenerate elliptic systems and some applications

Abstract

In this paper we develop a detailed study on maximum and comparison principles related to the following nonlinear eigenvalue problem{−Δpu=λa(x)|v|β1−1vinΩ;−Δqv=μb(x)|u|β2−1uinΩ;u=v=0on∂Ω, where p,q∈(1,∞), β1,β2>0 satisfy β1β2=(p−1)(q−1), Ω⊂Rn is a bounded domain with C2-boundary, a,b∈L∞(Ω) are given functions, both assumed to be strictly positive on compact subsets of Ω, and Δp and Δq are quasilinear elliptic operators, stand for p-Laplacian and q-Laplacian, respectively. We classify all couples (λ,μ)∈R2 such that both the (weak and strong) maximum and comparison principles corresponding to the above system hold in Ω. Explicit lower bounds for principal eigenvalues of this system in terms of the measure of Ω are also proved. As application, given λ,μ≥0 we measure explicitly how small has to be |Ω| so that weak and strong maximum principles associated to the above problem hold in Ω.