  We study the bifurcation of solutions of semilinear elliptic boundary value
problems of the form \begin{align*}
  \begin{aligned}
  -\Delta u &= f_\lambda(|x|,u,|\nabla u|) &&\text{in }\Omega,
  u &= 0 &&\text{on }\partial\Omega,
  \end{aligned} \end{align*} on an annulus $\Omega\subset\mathbb{R}^N$, with a
concave-convex nonlinearity, a special case being the nonlinearity first
considered by Ambrosetti, Brezis and Cerami: $f_\lambda(|x|,u,|\nabla
u|)=\lambda|u|^{q-2}u + |u|^{p-2}u$ with $1<q<2<p$. Although the trivial
solution $u_0\equiv0$ is nondegenerate if $\lambda=0$ we prove that
$(\lambda_0,u_0)=(0,0)$ is a bifurcation point. In fact, the bifurcation
scenario is very singular: We show that there are infinitely many global
continua of radial solutions
$\mathcal{C}_j^\pm\subset\mathbb{R}\times\mathcal{C}^1(\Omega)$,
$j\in\mathbb{N}_0$ which bifurcate from the trivial branch
$\mathbb{R}\times\{0\}$ at $(\lambda_0,u_0)=(0,0)$ and consist of solutions
having precisely $j$ nodal annuli. A detailed study of these continua shows
that they accumulate at $\mathbb{R}_{\ge0}\times\{0\}$ so that every
$(\lambda,0)$ with $\lambda\ge0$ is a bifurcation point. Moreover, adding a
point at infinity to $\mathcal{C}^1(\Omega)$ they also accumulate at
$\mathbb{R}\times\{\infty\}$, so there is bifurcation from infinity at every
$\lambda\in\mathbb{R}$.
