Existence and multiplicity of positive solutions for discrete anisotropic equations

Authors: MAREK GALEWSKI, RENATA WIETESKA

Abstract: In this paper we consider the Dirichlet problem for a discrete anisotropic equation with some function \alpha , a nonlinear term f, and a numerical parameter \lambda :\Delta (\alpha (k) |\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1)) + \lambda f(k,u(k))=0, k\in [1,T] . We derive the intervals of a numerical parameter \lambda for which the considered BVP has at least 1, exactly 1, or at least 2 positive solutions. Some useful discrete inequalities are also derived.

Keywords: Discrete boundary value problem, variational methods, Ekeland's variational principle, mountain pass theorem, Karush--Kuhn--Tucker theorem, positive solution, anisotropic problem

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