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Infinitely many solutions for a class of discrete non-linear boundary value problems. (English) Zbl 1176.39004

The following problem is considered: \[ -\Delta(\phi_p(\Delta u_{k-1})+ cx_{n-k})+ q_k\phi_p(u_k)=\lambda f(k, u_k),\quad k\in [1,N], \]
\[ u_0= u_{N+1}= 0, \] where \(f: [1, N]\times\mathbb{R}\to\mathbb{R}\) is a continuous function, \(\Delta u_{k-1}=u_k- u_{k-1}\) is the forward difference operator, \(q_k\in\mathbb{R}^+_0\) for all \(k\in[1, N]\), \(\phi_p(s):=|s|^{p-2}s\), \(1< p< +\infty\) and \(\lambda\in \mathbb{R}^+\).
Two types of results are given: the existence of either an unbounded sequence of solutions or a sequence of pairwise distinct non-zero solutions which converges to \(0\), depending on whether the nonlinear term has a suitable oscillating behavior, respectively, at infinity or at zero.

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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