Positive solutions for a class of p-Laplacian systems with multiple parameters

Consider the system − pu = λ1f (v)+ μ1h(u) in Ω, − qv = λ2g(u)+ μ2γ (v) in Ω, u = 0 = v on ∂Ω, where sz = div(|∇z|s−2∇z), s > 1, λ1, λ2, μ1 and μ2 are nonnegative parameters, and Ω is a bounded domain in RN with smooth boundary ∂Ω. We prove the existence of a large positive solution for λ1 + μ1 and λ2 +μ2 large when lim x→∞ f (M[g(x)]1/q−1) xp−1 = 0 for every M >0, limx→∞ h(x) xp−1 = 0 and limx→∞ γ (x) xq−1 = 0. In particular, we do not assume any sign conditions on f (0), g(0), h(0) or γ (0). We also discuss a multiplicity results when f (0) = g(0) = h(0) = γ (0) = 0. © 2007 Elsevier Inc. All rights reserved.