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

TL;DR

This paper proves the existence of multiple solutions for a nonlinear p-Laplacian boundary value problem on the Sierpinski gasket, involving a nonlocal term and nonlinear boundary conditions.

Contribution

It establishes the existence of two nontrivial weak solutions for a p-Laplacian equation on fractal domains with a specific nonlinear structure.

Findings

Existence of two nontrivial weak solutions.

Application of variational methods on fractal domain.

Analysis of nonlinear p-Laplacian on 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)\Delta_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 Sierpi\'nski gasket in $\mathbb{R}^2$, $\mathcal{S}_0$ is its boundary. $M : \mathbb{R} \to \mathbb{R}$ defined by $M(t) = at^k +b$ 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.