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,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta [\\varphi _{p}(\\Delta u(t-1))]=\\lambda a(t)\\varphi _{p}(u(t))+g(t,u(t), \\lambda ),\\quad t\\in [1,T+1]_{Z}, \\\\ \\Delta u(0)=u(T+2)=0, \\end{cases} $$\\end{document} where Delta u(t)=u(t+1)-u(t) is a forward difference operator, varphi _{p}(s)=|s|^{p-2}s (1< p<+infty ) is a one-dimensional p-Laplacian operator. λ is a positive real parameter, a: [1,T+1]_{Z}to [0,+infty ) and a(t_{0})>0 for some t_{0}in [1,T+1]_{Z}, g :[1,T+1]_{Z}times mathbb{R}^{2}to mathbb{R} satisfies the Carathéodory condition in the first two variables. We show that (lambda _{1},0) is a bifurcation point of the above problem, and there are two distinct unbounded continua mathscr{C}^{+} and mathscr{C}^{-}, consisting of the bifurcation branch mathscr{C} from (lambda _{1},0), where lambda _{1} 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, nin Z with m< n, [m, n]_{Z}:={m, m+1,ldots , n}.As the applications of the above result, we prove more details about the existence of constant sign solutions for the following problem: {−Δ[φp(Δu(t−1))]=λa(t)f(u(t)),t∈[1,T+1]Z,Δu(0)=u(T+2)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta [\\varphi _{p}(\\Delta u(t-1))]=\\lambda a(t)f(u(t)),\\quad t\\in [1,T+1]_{Z}, \\\\ \\Delta u(0)=u(T+2)=0, \\end{cases} $$\\end{document} where fin C(mathbb{R},mathbb{R}) with sf(s)>0 for sneq 0.