Global bifurcation of positive solutions for a superlinear p -Laplacian system

Abstract

We are concerned with the principal eigenvalue of (P) { − Δ p u = λ θ 1 φ p ( v ) , x ∈ Ω , − Δ p v = λ θ 2 φ p ( u ) , x ∈ Ω , u = 0 = v ,   x ∈ ∂ Ω and the global structure of positive solutions for the system (Q) { − Δ p u = λ f ( v ) , x ∈ Ω , − Δ p v = λ g ( u ) , x ∈ Ω , u = 0 = v ,   x ∈ ∂ Ω , where φ p ( s ) = | s | p − 2 s , Δ p s = div ( | ∇ s | p − 2 ∇ s ) , λ > 0 is a parameter, Ω ⊂ R N , N > 2 , is a bounded domain with smooth boundary ∂ Ω , f , g : R → ( 0 , ∞ ) are continuous functions with p -superlinear growth at infinity. We obtain the principal eigenvalue of ( P ) by using a nonlinear Krein–Rutman theorem and the unbounded branch of positive solutions for ( Q ) via bifurcation technology.