A class of semilinear elliptic equations on the Sierpinski gasket

ABSTRACT

We study the existence of a weak (strong) solution for the semilinear elliptic problem $\begin{array}{rl}-\mathrm{\Delta }u\left(x\right)-\lambda g\left(x\right)u\left(x\right)& =-f(x,u(x\left)\right)\mathrm{i}\mathrm{n}V\setminus V_0\\ u& =0\mathrm{o}\mathrm{n}{V}_0,\end{array}$ where V is the Sierpiński gasket in ${\mathbb{R}}^{N-1}(N\ge 2)$, $V_0$ is its boundary (consisting of its N corners) and λ is a real parameter. Here $g:V\to \mathbb{R}$ and $f:V\times \mathbb{R}\to \mathbb{R}$ are functions satisfying suitable hypotheses.

Keywords:

• Sierpinski gasket
• semilinear elliptic equations
• fractal domains
• monotone operators

AMS CLASSIFICATIONS:

• 35J60
• 28A80
• 47H05

Disclosure statement

No potential conflict of interest was reported by the authors.