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

• Main Author: Fumei Ye
• Format: Article
• Language: English
• Published: SpringerOpen 2021-05-01
• Series: Advances in Difference Equations
• Subjects: Discrete p-Laplacian; Principal eigenvalue; Unilateral global bifurcation; Constant sign solutions
• Online Access: https://doi.org/10.1186/s13662-021-03309-9

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 < p < + ∞ $1< p<+\infty $ ) is a one-dimensional p-Laplacian operator. λ is a positive real parameter, a : [ 1 , T + 1 ] Z → [ 0 , + ∞ ) $a: [1,T+1]_{Z}\to [0,+\infty )$ and a ( t 0 ) > 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 < n $m< n$ , [ m , n ] Z : = { m , m + 1 , … , 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 [ φ p ( Δ u ( t − 1 ) ) ] = λ a ( t ) f ( u ( t ) ) , t ∈ [ 1 , T + 1 ] Z , Δ u ( 0 ) = u ( T + 2 ) = 0 , $$ \textstyle\begin{cases} -\Delta [\varphi _{p}(\Delta u(t-1))]=\lambda a(t)f(u(t)),\quad t\in [1,T+1]_{Z}, \\ \Delta u(0)=u(T+2)=0, \end{cases} $$ where f ∈ C ( R , R ) $f\in C(\mathbb{R},\mathbb{R})$ with s f ( s ) > 0 $sf(s)>0$ for s ≠ 0 $s\neq 0$ .