Nonexistence of Solutions and an Anti‐Maximum Principle for Cooperative Systems with the p ‐Laplacian

We obtain a nonexistence result and an anti‐maximum principle for weak solutions u = (u 1 ,…, u n ) to the following strictly cooperative elliptic system,Here, Ω C IR N is a bounded domain with a C 2,α ‐ boundary δΩ, for some α ∈ (0,1), Δ p denotes the p ‐Laplacian defined by Δ p u = div (|∇u| p‐2 ∇u) for p 6 (1,∞), and the coefficients a ij ( 1 ≤ i,j ≤ n) are assumed to be constants satisfying a ij > 0 for i ≠ j (a strictly cooperative system). We assume 0 ≤ f i , ∈ L ∞ (Ω) (1 ≤ i ≤ n). For ∇ + = ∇ ‐ ∇ ‐ = ∈ IR and f = ( f 1 ,…, f n ) 0 in Ω, let μ 1 denote the first eigenvalue of the (p ‐ 1)‐homogeneous system (S). Assuming f ≠ 0 in Ω, we show: (i) if ∧ + = ∧ ‐ = μ 1 , then ( S ) has no solution; and (ii) if ∧ + , ∧ ‐ ∧ (μ 1 ,μ 1 + δ), for some δ > 0 small enough, then u i < 0 in Ω and δ ui /δ v > 0 on δΩ (1 ≤ i ≤ n). Our methods for the system ( S ) are completely different from the case n = 1 (a single equation). For n ≥ 2 and p ≥ 2, mild additional hypotheses are imposed on the domain Ω and the matrix.