An application of a discrete Sobolev inequality to discretised Kirchhoff equations

Abstract

We develop a discrete Sobolev inequality, and we use this inequality to analyse the nonlocal discretised Kirchhoff equation$-A\left(\right(a⁎(g\circ |\Delta u\left|\right)\left)\right(b+1\left)\right)\left({\Delta }^{2}u\right)\left(n\right)=\lambda f(n,u(n+1\left)\right)\text{, }n\in \{0,1,2,\dots ,b\},$ where ⁎ represents a finite convolution. The equation is a discrete analogue of the classical steady-state Kirchhoff equation in one space dimension. Existence of at least one positive solution is investigated under the assumption that the equation is subject to two-point boundary data such that $u\left(0\right)=0$. Thus, both Dirichlet and right-focal data are captured by our results. An interesting aspect of our theory is that the coefficient function A may be both vanishing and sign-changing.