EIGENVALUES AND FUŠCIK-SPECTRUM OF THE RADIALLY SYMMETRIC $p$-LAPLACIAN

WOLFGANG REICHEL

reichel@math.unibas.ch

Mathematisches Institut
Universität Basel
Rheinsprung 21, CH-4051 Basel, Switzerland

For $p>1$ we consider the $p$-Laplacian boundary value problem

$${\rm div}(\vert\nabla u\vert^{p-2}\nabla u)+(-q+\lambda w)\vert u\vert^{p-2} u = 0 B_1(0)\subset\mathbbm{R}^n$$ (1)

with homogeneous boundary conditions on $\partial B_1(0)$. Radially symmetric solutions satisfy an ordinary differential equation. We will discuss analytical and numerical tools to find all radial eigenvalues of the problem. In generalization to (1) we also discuss the boundary value problem

$${\rm div}(\vert\nabla u^{p-2}\vert\nabla u)-q\vert u\vert^{p-2} u +w\Big(\mu (u^+)^{p-1} - \nu (u^-)^{p-1}\Big) = 0 B_1(0)$$ (2)

with homogeneous boundary conditions, where $u^+(x)=\max\{u(x),0\}$ and $u(x)=u^+(x)-u^-(x)$. A pair of constants $(\mu,\nu)\in\mathbbm{R}^2$ for which a non-trivial solution $u$ exists is called a Fucik-eigenvalue; the collection of all Fucik-eigenvalues is called the Fucik-spectrum. We describe the entire radial Fucik-spectrum analytically, and we give an algorithm for its computation. Numerical result will be presented.