Parametric critical point theorems and their applications to boundary value problems on the Sierpi\'{n}ski Gasket

TL;DR

This paper extends classical variational methods with parameters to analyze critical points and their behavior, applying these results to boundary value problems on the Sierpiński Gasket.

Contribution

It introduces parametric versions of critical point theorems and explores their application to boundary value problems on fractal domains.

Findings

Behavior of critical points under parameter convergence

Existence of solutions for boundary value problems on the Sierpiński Gasket

Conditions ensuring mountain pass geometry

Abstract

In this note we consider the classical variational tools like: Ekelenad's Variational Principle, Mountain Pass Lemma and some of their corollaries subject to a parameter. Next, we investigate the behaviour of critical points obtained once a sequence of parameters is allowed to be convergent. Applications for the Dirichlet Boundary Value Problem on the Sierpi\'{n}ski Gasket are given in presence of assumptions which lead to fulfillment of the mountain geometry.