On the existence of three solutions for the Dirichlet problem on the Sierpinski gasket

TL;DR

This paper proves the existence of at least three solutions for certain Dirichlet problems on the Sierpinski gasket using critical point theorems, including Ricceri's theorem and mountain pass techniques.

Contribution

It introduces new existence results for multiple solutions of Dirichlet problems on fractals, applying advanced critical point theorems to the Sierpinski gasket.

Findings

At least three solutions exist for specific two-parameter Dirichlet problems.

Existence of three nonzero solutions for perturbed problems is established.

Utilizes Ricceri's three critical points theorem and mountain pass theorems.

Abstract

We apply a recently obtained three critical points theorem of B. Ricceri to prove the existence of at least three solutions of certain two-parameters Dirichlet problems defined on the Sierpinski gasket. We also show the existence of at least three nonzero solutions of certain perturbed two-parameters Dirichlet problems on the Sierpinski gasket, using both the mountain pass theorem of Ambrosetti-Rabinowitz and that of Pucci-Serrin.