Existence of solutions for a non-local equation on the Sierpinski gasket

Abstract

This paper presents results on a class of Kirchhoff problems, using the theory established in a previous work \cite{Ricceri01}. Specifically, we focus on the Sierpi\'{n}ski gasket $(V,\|\cdot\|)$ in $(\RR^{n-1},|\cdot|)$, where $n\geq 2$, and $V_0$ is its intrinsic boundary consisting of $n$ corners. We consider a continuous function $f:\RR\to \RR$ and show that the following two statements are equivalent:\\ (i) the restriction of $f$ to $\left[-\frac{(2n+3)\sqrt{2\pi}}{4}, \frac{(2n+3)\sqrt{2\pi}}{4}\right]$ is not constant, and \\ (ii) for every convex set $\Theta$ that is dense in $C^0(V)\times C^0(V)$, there exists $(\alpha,\beta)\in \Theta$ such that a class of Kirchhoff problems, as follows:
\begin{equation*}
\left\{\begin{array}{ll}
-\tan\left(\|u\|^2\right)\Delta u+t^2u(t)=\beta(t)f(u(t))+\alpha(t),\quad
t\in {V\setminus V_0}\\
u(t)=0,\quad t\in V_0,\\
\|u\|^2<\frac{\pi}{2}
\end{array}\right.
\end{equation*}
has at least two classical solutions.