Existence results for nonlinear elliptic problems on fractal domains

TL;DR

This paper establishes the existence of solutions for a nonlinear elliptic boundary value problem on the Sierpiński fractal, identifying specific eigenvalues where solutions are guaranteed using critical point theory.

Contribution

It provides the first existence results for nonlinear elliptic problems on fractal domains, specifically on the Sierpiński fractal, using variational methods.

Findings

Existence of at least one non-trivial weak solution for certain eigenvalues.

Identification of an open interval of positive eigenvalues where solutions exist.

Application of critical point theory to fractal domain problems.

Abstract

Some existence results for a parametric Dirichlet problem defined on the Sierpi\'nski fractal are proved. More precisely, a critical point result for differentiable functionals is exploited in order to prove the existence of a well determined open interval of positive eigenvalues for which the problem admits at least one non-trivial weak solution.