Global bifurcation and constant sign solutions of discrete boundary value problem involving p-Laplacian

Abstract

We study the unilateral global bifurcation result for the one-dimensional discrete p-Laplacian problem {−Δ[φp(Δu(t−1))]=λa(t)φp(u(t))+g(t,u(t),λ),t∈[1,T+1]Z,Δu(0)=u(T+2)=0, where Δ u(t) = u(t+ 1 ) − u(t) is a forward difference operator, φp(s) = | s| p−2s (1 < p< + ∞ ) is a one-dimensional p-Laplacian operator. λ is a positive real parameter, a: [ 1 , T+ 1 ] Z→ [ 0 , + ∞ ) and a(t) > 0 for some t∈ [ 1 , T+ 1 ] Z, g: [ 1 , T+ 1 ] Z× R2→ R satisfies the Carathéodory condition in the first two variables. We show that (λ1, 0 ) is a bifurcation point of the above problem, and there are two distinct unbounded continua C+ and C−, consisting of the bifurcation branch C from (λ1, 0 ) , where λ1 is the principal eigenvalue of the eigenvalue problem corresponding to the above problem. Let T> 1 be an integer, Z denote the integer set for m, n∈ Z with m 0 for s≠ 0.