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

TL;DR

This paper investigates an elliptic boundary value problem with concave-convex nonlinearity, revealing infinitely many global bifurcation branches from a trivial solution, with solutions characterized by their number of nodal annuli and bifurcations occurring at multiple parameter values.

Contribution

It demonstrates the existence of infinitely many global continua bifurcating from a single trivial solution in a nonlinear elliptic problem with complex bifurcation structure.

Findings

Infinitely many global bifurcation branches from the trivial solution.

Bifurcation points occur at every non-negative parameter value.

Solutions are characterized by the number of nodal annuli.

Abstract

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 $\lambda\in\mathbb{R}$.