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

Abstract

AbstractWe study the unilateral global bifurcation result for the one-dimensional discrete p-Laplacian problem $$ \textstyle\begin{cases} -\Delta [\varphi _{p}(\Delta u(t-1))]=\lambda a(t)\varphi _{p}(u(t))+g(t,u(t), \lambda ),\quad t\in [1,T+1]_{Z}, \\ \Delta u(0)=u(T+2)=0, \end{cases} $$ { − 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 , where $\Delta u(t)=u(t+1)-u(t)$ Δ u ( t ) = u ( t + 1 ) − u ( t ) is a forward difference operator, $\varphi _{p}(s)=|s|^{p-2}s$ φ p ( s ) = | s | p − 2 s ($1< p<+\infty $ 1 < p < + ∞ ) is a one-dimensional p-Laplacian operator. λ is a positive real parameter, $a: [1,T+1]_{Z}\to [0,+\infty )$ a : [ 1 , T + 1 ] Z → [ 0 , + ∞ ) and $a(t_{0})>0$ a ( t 0 ) > 0 for some $t_{0}\in [1,T+1]_{Z}$ t 0 ∈ [ 1 , T + 1 ] Z , $g :[1,T+1]_{Z}\times \mathbb{R}^{2}\to \mathbb{R}$ g : [ 1 , T + 1 ] Z × R 2 → R satisfies the Carathéodory condition in the first two variables. We show that $(\lambda _{1},0)$ ( λ 1 , 0 ) is a bifurcation point of the above problem, and there are two distinct unbounded continua $\mathscr{C}^{+}$ C + and $\mathscr{C}^{-}$ C − , consisting of the bifurcation branch $\mathscr{C}$ C from $(\lambda _{1},0)$ ( λ 1 , 0 ) , where $\lambda _{1}$ λ 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 $m, n\in Z$ m , n ∈ Z with $m< n$ m < n , $[m, n]_{Z}:=\{m, m+1,\ldots , n\}$ [ m , n ] Z : = { m , m + 1 , … , n } .As the applications of the above result, we prove more details about the existence of constant sign solutions for the following problem: $$ \textstyle\begin{cases} -\Delta [\varphi _{p}(\Delta u(t-1))]=\lambda a(t)f(u(t)),\quad t\in [1,T+1]_{Z}, \\ \Delta u(0)=u(T+2)=0, \end{cases} $$ { − Δ [ φ p ( Δ u ( t − 1 ) ) ] = λ a ( t ) f ( u ( t ) ) , t ∈ [ 1 , T + 1 ] Z , Δ u ( 0 ) = u ( T + 2 ) = 0 , where $f\in C(\mathbb{R},\mathbb{R})$ f ∈ C ( R , R ) with $sf(s)>0$ s f ( s ) > 0 for $s\neq 0$ s ≠ 0 .