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.


39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] DOI: 10.1016/j.jmaa.2004.10.055 · Zbl 1074.39022
[2] Jiang D, Math. Inequal. Appl. 7 pp 523– (2004)
[3] DOI: 10.1016/S0898-1221(02)00095-0 · Zbl 1005.39014
[4] DOI: 10.1016/j.jmaa.2006.02.091 · Zbl 1113.39018
[5] DOI: 10.1016/j.na.2008.04.021 · Zbl 1166.39006
[6] DOI: 10.1016/j.camwa.2008.01.025 · Zbl 1155.39301
[7] DOI: 10.1007/s00010-006-2855-5 · Zbl 1128.39010
[8] DOI: 10.1155/2008/345916 · Zbl 1146.39028
[9] DOI: 10.1016/j.jmaa.2006.03.017 · Zbl 1112.39008
[10] DOI: 10.1016/j.jmaa.2005.12.047 · Zbl 1167.39305
[11] DOI: 10.1155/ADE.2005.93 · Zbl 1098.39001
[12] DOI: 10.1016/j.na.2003.11.012 · Zbl 1070.39005
[13] Bonanno G, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities pp 1– (2009) · Zbl 1177.34038
[14] DOI: 10.1006/jdeq.2001.4092 · Zbl 1013.49001
[15] DOI: 10.1016/S0377-0427(99)00269-1 · Zbl 0946.49001
[16] DOI: 10.1016/j.jde.2008.02.025 · Zbl 1149.49007
[17] Agarwal RP, Difference Equations and Inequalities: Theory, Methods and Applications (2000)
[18] Agarwal RP, Positive Solutions of Differential, Difference and Integral Equations (1999)
[19] Kelly WG, Difference Equations, An Introduction with Applications (1991)
[20] Rabinowitz, PH. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conferences in Mathematics. Vol. 65, Providence, RI: American Mathematical Society.
[21] Struwe M, Variational Methods (1996)
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