A note on gradient-type systems on fractals. (February 2015) - Breckner, Brigitte E.; Varga, Csaba
Abstract: The purpose of the paper is to point out that well-established methods to study the multiplicity of solutions of nonlinear elliptic problems defined on open sets in Euclidean spaces can be also used in the case of problems defined on fractals. More exactly, using variational methods and an abstract critical-point theorem by B. Ricceri, we prove the existence of three nonzero weak solutions of certain gradient-type systems defined on a famous fractal, the Sierpinski gasket. The paper emphasizes the way we overcame the difficulties arising from the major structural differences between the highly non-smooth Sierpinski gasket and the open subsets of Euclidean spaces on which gradient-type systems are usually considered. Highlights: We apply a Ricceri-type critical-point theorem to elliptic systems on fractals. We emphasize the special structure of the Sierpinski gasket. We overcame difficulties arising from the non-smooth structure of fractals.