Existence of Multiple Solutions of a Kirchhoff Type $p$-Laplacian Equation on the Sierpiński Gasket

Abstract

In this paper, we study the following boundary value problem involving the weak $p$-Laplacian.

$$ -M\bigl(\|u\|_{\mathcal{E}_{p}}^{p}\bigr)\Delta _{p} u = h(x,u) \quad \text{in}\ \mathcal{S}\setminus \mathcal{S}_{0}; \quad u = 0 \ \text{on}\ \mathcal{S}_{0}, $$

where $\mathcal{S}$ is the Sierpiński gasket in $\mathbb{R}^{2}$, $\mathcal{S}_{0}$ is its boundary, $M: \mathbb{R}^{+} \to \mathbb{R}$ is defined by $M(t) = at^{k} +b$, where $a,b,k >0$ and $h: \mathcal{S}\times \mathbb{R}\to \mathbb{R}$. We will show the existence of two nontrivial weak solutions to the above problem.