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Existence of solutions for a nonlinear discrete system involving the \(p\)-Laplacian. (English) Zbl 1249.39009

The authors consider the discrete boundary value problem \[ \Delta \bigl (\Phi _p(\Delta u(t-1))\bigr )+\lambda \nabla F(t,u(t))=0,\quad t\in [1,M]_{\mathbb Z},\; u(0)=0=u(M+1), \] where \(\Phi _p(s)=| s| ^{p-1}s\), \(\lambda >0\) is a parameter and \(F\:[0,M]_{\mathbb Z}\times {\mathbb R}^m\to {\mathbb R}\) is continuously differentiable. Using the critical point theory, various existence statements for solvability of the above problem are proved. A typical result is the following theorem:
Suppose that \(F(0,0)=0\) and there exists a number \(\beta >p\) such that \[ \limsup _{| x| \to \infty } \frac {F(t,x)}{| x| ^{\beta }}>0\quad \text{for all }t\in [0,M]_{\mathbb Z}. \] If \(\lambda >0\), then the above problem has at least one solution.

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
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