FUŠCSIK SPECTRUM OF THE LAPLACIAN ON SOME SUBDOMAINS OF THE PLANE

JIŠRSI HORÁK

horak@math.unibas.ch

WOLFGANG REICHEL

reichel@math.unibas.ch

Mathematisches Institut
Universität Basel
Rheinsprung 21, CH-4057, Basel, Switzerland

The problem $\Delta u + \mu u^+ - \nu u^- = 0$ in $\Omega\subset\mathbb{R}^2$ with $u=0$ on $\partial\Omega$ is studied, where $u^\pm=\max(0,\pm u)$, $\mu,\nu\in\mathbb{R}$. A pair $(\mu,\nu)$ is called a Fucík eigenvalue if there exists a function $u_{\mu,\nu}\neq 0$ that solves the problem. Local existence results are obtained and a numerical method is used to approximate some of the Fucík eigenvalues on various domains in $\mathbb{R}^2$. Based on a variational formulation in $W^{1,2}_0(\Omega)$, critical points of the functional $F(u)=\int_\Omega \vert\nabla u\vert^2$d$x$ subject to constraints $\int_\Omega (u^+)^2$d$x=t$ and $\int_\Omega (u^-)^2$d$x=1-t$ with $t\in(0,1)$ are sought. The information obtained numerically is compared to the known analytical results.