Existence of multiple solutions of a p-Laplacian equation on the Sierpinski Gasket

Abstract

In this paper we study the following boundary value problem involving the weak p-Laplacian. \begin{equation*} \quad -M(\|u\|_{\mathcal{E}_p}^p)Δ_p u = h(x,u) \; \text{in}\; \mathcal{S}\setminus\mathcal{S}_0; \quad u = 0 \; \mbox{on}\; \mathcal{S}_0, \end{equation*} where $\mathcal{S}$ is the Sierpiński gasket in ${\mathbb{R}}^2$, ${\mathcal{S}}_0$ is its boundary. $M:\mathbb{R}\to \mathbb{R}$ defined by $M(t)=a{t}^k+b$ and $a,b,k>0$ and $h:\mathcal{S}\times \mathbb{R}\to \mathbb{R}.$ We will show the existence of two nontrivial weak solutions to the above problem.

18 pages. arXiv admin note: text overlap with arXiv:1807.06881