Nonlinear Elliptic Equations on the Sierpiński Carpet

Abstract

This paper deals with a semilinear elliptic equation for a properly defined Laplace operator on the Sierpiński carpet. We investigate the existence of solutions, under Dirichlet boundary conditions, for the problem
∆u+a(x)u=f(x,u).
This work is motivated by previos studies established for such problems on the Sierpiński gasket by many authors: Kenneth J. Falconer, Jiaxin Hu, G.Bonanno and others (more details can be found in Introduction). The approach used in this study is based on variational methods, more specifically, the the mountain pass theorem of Ambrosetti and Rabinowit, and Sobolev type inequality, which was of principal use for achieving other results.