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Global structure of sign-changing solutions for discrete Dirichlet problems by Wei, Liping ;Ma, Ruyun is a Mathematics article available to read on EtoBox.

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Abstract Let T\gt 1 be an integer, {\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 \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 \lambda \gt 0 is a parameter, f\in C({\mathbb{R}},{\mathbb{R}}) satisfies f(0)=0,\hspace{.1em}sf(s)\gt 0 for all s\ne 0 and 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 \lambda lying in various intervals in {\mathbb{R}} . Moreover, we point out that these solutions change their sign exactly k-1 times, where k\in \{1,2,\ldots ,T\} .

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Author

Wei, Liping ;Ma, Ruyun

Publisher

Walter de Gruyter GmbH

Published

2020

Language

EN

Field

Mathematics (Physical Sciences)