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Existence of solutions for discrete boundary value problems with second order dependence on parameters. (English) Zbl 1463.39005

The authors consider the discrete boundary value problem \[ -\Delta(a(k-1,\Delta u(k-1)))=\delta(k)f(k,u(k),w),\quad \Delta u(0)=\Delta u(N)=0. \] The difference operator can be seen as a discrete counterpart of an anisotropic operator. Using the mountain pass lemma, the existence of a nontrivial solution under some hypotheses is proved. Under additional conditions, also the uniqueness of the solution is proved.
Reviewer: Pavel Rehak (Brno)

MSC:

39A12 Discrete version of topics in analysis
35B38 Critical points of functionals in context of PDEs (e.g., energy functionals)
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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