## 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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### References:

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