On sequences of large solutions for discrete anisotropic equations

Abstract

In this paper, we determine a concrete interval of positive parameters $\lambda$, for which we prove the existence of infinitely many solutions for an anisotropic discrete Dirichlet problem \begin{align*} -\Delta\left( \alpha\left( k\right) |\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1)\right) =\lambda f(k,u(k)),\quad k\in \mathbb{Z} \lbrack1,T], \end{align*} where the nonlinear term $f: \mathbb{Z} \lbrack1,T]\times \mathbb{R}\rightarrow\mathbb{R}$ has an appropriate behavior at infinity, without any symmetry assumptions. The approach is based on critical point theory.

Keywords

• discrete nonlinear boundary value problem; variational methods
• anisotropic problem; infinitely many solutions.