Multiple Solutions to a Dirichlet Problem on the Sierpinski Gasket

Abstract

We investigate the existence of at least two nontrivial solutions to a Dirichlet problem on the Sierpinski gasket. We develop some general abstract multiplicity theorem which we apply to problem under consideration. Our approach relies on the fact that the action functional is a difference of two continuously differentiable convex functionals and therefore we can apply the ideas related to the Fenchel-Young conjugacy together to get one critical point together with the mountain pass geometry to get the other one.