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

Abstract

Abstract We study the unilateral global bifurcation result for the one-dimensional discrete p-Laplacian problem { − Δ [ φ p ( Δ u ( t − 1 ) ) ] = λ a ( t ) φ p ( u ( t ) ) + g ( t , u ( t ) , λ ) , t ∈ [ 1 , T + 1 ] Z , Δ u ( 0 ) = u ( T + 2 ) = 0 , $$ \textstyle\begin{cases} -\Delta [\varphi _{p}(\Delta u(t-1))]=\lambda a(t)\varphi _{p}(u(t))+g(t,u(t), \lambda ),\quad t\in [1,T+1]_{Z}, \\ \Delta u(0)=u(T+2)=0, \end{cases} $$ where Δ u ( t ) = u ( t + 1 ) − u ( t ) $\Delta u(t)=u(t+1)-u(t)$ is a forward difference operator, φ p ( s ) = | s | p − 2 s $\varphi _{p}(s)=|s|^{p-2}s$ ( 1 0 $a(t_{0})>0$ for some t 0 ∈ [ 1 , T + 1 ] Z $t_{0}\in [1,T+1]_{Z}$ , g : [ 1 , T + 1 ] Z × R 2 → R $g :[1,T+1]_{Z}\times \mathbb{R}^{2}\to \mathbb{R}$ satisfies the Carathéodory condition in the first two variables. We show that ( λ 1 , 0 ) $(\lambda _{1},0)$ is a bifurcation point of the above problem, and there are two distinct unbounded continua C + $\mathscr{C}^{+}$ and C − $\mathscr{C}^{-}$ , consisting of the bifurcation branch C $\mathscr{C}$ from ( λ 1 , 0 ) $(\lambda _{1},0)$ , where λ 1 $\lambda _{1}$ is the principal eigenvalue of the eigenvalue problem corresponding to the above problem. Let T > 1 $T>1$ be an integer, Z denote the integer set for m , n ∈ Z $m, n\in Z$ with m 0 $sf(s)>0$ for s ≠ 0 $s\neq 0$ .

Keywords

• Discrete p-Laplacian
• Principal eigenvalue
• Unilateral global bifurcation
• Constant sign solutions