Three solutions to discrete anisotropic problems with two parameters (Q403165)

An extension of the three critical point theorem is provided. Some known three critical point theorems are recalled in Section 3 and the main result is established in Section 4. In Section 5 it is applied to a proof of existence of at least two nontrivial solutions of the discrete boundary value problem \[ \begin{gathered}-\Delta\big[|\Delta x(k-1)|^{p(k-1)-2}\,\Delta x(k-1)\big]+\gamma\,g(k,x(k))+\lambda\,f(k,x(k))=0,\qquad k\in\{1,\dots,T\},\\ x(0)=0=x(T+1),\end{gathered} \] where \(\gamma,\lambda>0\) are numerical parameters, \(f,g:\{1,\dots,T\}\times\mathbb{R}\to\mathbb{R}\) are continuous functions satisfying some additional assumptions, and \(p:\{0,\dots,T+1\}\to\mathbb{R}_+\). A variational framework for the latter problem is given in Section 2.