BIFURCATIONS FROM INTERVALS FOR DISCRETE BOUNDARY VALUE PROBLEMS INVOLVING p-LAPLACIANS

We are concerned with the global bifurcation results for p-Laplacian discrete problem −Δ[φp(Δu(t−1))]=λa(t)φp(u(t))+a(t)f(t,u(t),λ)+g(t,u(t),λ),t∈[1,T]Z,u(0)=u(T+1)=0, where λ>0 is a parameter, a:[1,T]Z→[0,∞),f,g∈C([0,T+1]Z×ℝ2,ℝ), Δu(t)=u(t+1)−u(t) is the forward difference operator, φp(s)=|s|p−2s(1<p<+∞). We shall show that there are two distinct unbounded continua 𝒞+ and 𝒞−, consisting of the bifurcation branch 𝒞 if f is not necessarily differentiable at the origin with respect to φp(u), and there are two distinct unbounded continua 𝒟+ and 𝒟−, consisting of the bifurcation branch 𝒟 if f is not necessarily differentiable at infinity with respect to φp(u). As the applications of the above result, we shall obtain that there exist at least a positive solution and a negative one for the half-quasilinear problem −Δ[φp(Δu(t−1))]=μa(t)F(u(t))+α(t)φp(u+(t))+β(t)φp(u−(t)),t∈[1,T]Z,u(0)=u(T+1)=0, where μ≠0 is a parameter, a:[1,T]Z→(0,+∞),α,β:[1,T]Z→ℝ,u+=max{u,0},u−=−min{u,0}, F∈C(ℝ,ℝ) satisfies sF(s)>0.