A pathological example in Nonlinear Spectral Theory

We construct an open set $\mathrm{\Omega }\subset {\mathbb{R}}^N$ on which an eigenvalue problem for the $p-$Laplacian has not isolated first eigenvalue and the spectrum is not discrete. The same example shows that the usual Lusternik-Schnirelmann minimax construction does not exhaust the whole spectrum of this eigenvalue problem.