Infinitely many global continua bifurcating from a single solution of an elliptic problem with concave-convex nonlinearity

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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$, 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 $u_0\equiv 0$ 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 ${\mathcal{C}}_j^{\pm }\subset \mathbb{R}\times {\mathcal{C}}^1(\mathrm{\Omega })$, $j\in {\mathbb{N}}_0$ 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 ${\mathbb{R}}_{\ge 0}\times \{0\}$ so that every $(\lambda ,0)$ with $\lambda \ge 0$ is a bifurcation point. Moreover, adding a point at infinity to ${\mathcal{C}}^1(\mathrm{\Omega })$ they also accumulate at $\mathbb{R}\times \{\mathrm{\infty }\}$, so there is bifurcation from infinity at every $\lambda \in \mathbb{R}$.