The multiplicity of solutions for nonlinear difference equations involving the p -Laplacian is investigated. The approach is based on critical point theorems in the setting of finite dimensional Banach spaces.

Nonlinear difference equations investigated via critical point methods

Abstract In this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.

Variational Methods on Finite Dimensional Banach Spaces and Discrete Problems

We investigate the existence of at least three solutions for a discrete nonlinear Neumann boundary value problem involving the -Laplacian. Our approach is based on three critical points theorems.

https://link.springer.com/content/pdf/10.1155%2F2010%2F862016.pdf

Three Solutions for a Discrete Nonlinear Neumann Problem Involving the -Laplacian

A general min-max principle established by Ghoussoub is extended to the case of functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function. Some topological properties of the min-max-generated critical points in such a framework are then pointed out.

/pdf/existence-and-classification-of-critical-points-for-3hi7ueztjs.pdf

Existence and classification of critical points for nondifferentiable functions

Given a bounded below, lower semi-continuous function f on an infinite dimensional Banach space or a non-compact manifold X , we consider various possibilities of perturbing f by an element p of a reasonable class of functions A in such a way that for the new functional f- p , the minimization problem inf X (f- p ) is well-posed (i.e., every minimizing sequence is convergent).

Chapter 10 Perturbed minimization principles and applications

We extend the min-max methods used in the critical point theory of differentiable functionals on smooth manifolds to the case of continuous functionals on a complete metric space. We study the topological properties of the min-max generated critical points in this new setting by adopting the methodology developed by Ghoussoub in the smooth case. Many old and new results are extended and unified and some applications are given.

On the Existence and the Classification of Critical Points for Non-Smooth Functionals

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On the Existence and the Classification of Critical Points for Non-Smooth Functionals