Global structure of sign-changing solutions for discrete Dirichlet problems

Abstract

Abstract Let T > 1 T\gt 1 be an integer, T ≔ [ 1 , T ] Z = { 1 , 2 , … , T } , T ˆ ≔ { 0 , 1 , … , T + 1 } {\mathbb{T}}:= {{[}1,T]}_{{\mathbb{Z}}}=\{1,2,\ldots ,T\},\hspace{.0em}\hat{{\mathbb{T}}}:= \{0,1,\ldots ,T+1\} . In this article, we are concerned with the global structure of the set of sign-changing solutions of the discrete second-order boundary value problem { Δ 2 u ( x − 1 ) + λ h ( x ) f ( u ( x ) ) = 0 , x ∈ T , u ( 0 ) = u ( T + 1 ) = 0 , \left\{\begin{array}{l}{\mathrm{\Delta}}^{2}u(x-1)+\lambda h(x)f(u(x))=0,\hspace{1em}x\in {\mathbb{T}},\\ u(0)=u(T+1)=0,\end{array}\right. where λ > 0 \lambda \gt 0 is a parameter, f ∈ C ( ℝ , ℝ ) f\in C({\mathbb{R}},{\mathbb{R}}) satisfies f ( 0 ) = 0 , s f ( s ) > 0 f(0)=0,\hspace{.1em}sf(s)\gt 0 for all s ≠ 0 s\ne 0 and h : T ˆ → [ 0 , + ∞ ) h:\hat{{\mathbb{T}}}\to {[}0,+\infty ) . By using the directions of a bifurcation, we obtain existence and multiplicity of sign-changing solutions of the above problem for λ \lambda lying in various intervals in ℝ {\mathbb{R}} . Moreover, we point out that these solutions change their sign exactly k − 1 k-1 times, where k ∈ { 1 , 2 , … , T } k\in \{1,2,\ldots ,T\} .