Existence of Multiple Weak Solutions for Nonlinear Problems on the Sierpiński Gasket

Abstract

In this paper, we study the existence of at least two, three and infinitely many solutions for nonlinear problems on the Sierpiński gasket, modelling some physical phenomena such as reaction-diffusion problems, elastic properties of fractal media and flow through fractal non-smooth domains. We will obtain the existence of two weak solutions when nonlinear term f(x,t) is non-negative, when it is non-positive in neighbored of zero and otherwise is positive we will show the existence of three weak solutions, and when it is odd we will get the existence of infinitely many solutions. The results are proved by using some critical point theorems.

Keywords

Sierpiński gasket / fractal domains / nonlinear elliptic equation / variational methods / critical point