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

Abstract

AbstractWe obtain a nonexistence result and an anti‐maximum principle for weak solutions u = (u1,…, un) to the following strictly cooperative elliptic system, equation image Here, Ω C IRN is a bounded domain with a C2,α ‐ boundary δΩ, for some α ∈ (0,1), Δp denotes the p‐Laplacian defined by Δpu = div (|∇u|p‐2∇u) for p 6 (1,∞), and the coefficients aij (1 ≤ i,j ≤ n) are assumed to be constants satisfying aij > 0 for i ≠ j (a strictly cooperative system). We assume 0 ≤ fi, ∈ L∞(Ω) (1 ≤ i ≤ n). For ∇+ = ∇‐ ∇‐ = ∈ IR and f = (f1,…,fn) 0 in Ω, let μ1 denote the first eigenvalue of the (p ‐ 1)‐homogeneous system (S). Assuming f ≠ 0 in Ω, we show: (i) if ∧+ = ∧‐ = μ1, then (S) has no solution; and (ii) if ∧+, ∧‐ ∧ (μ1,μ1 + δ), for some δ > 0 small enough, then ui < 0 in Ω and δui/δv > 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 equation image .