Intervals of bifurcation points for semilinear elliptic problems

TL;DR

This paper investigates the structure and behavior of solution continua for semilinear elliptic problems with nonlinearities having multiple zeros, revealing complex bifurcation phenomena and conditions where all positive parameters are bifurcation points.

Contribution

It provides new insights into the bifurcation structure of solutions when the nonlinearity has multiple zeros, including cases with infinitely many zeros and unusual bifurcation properties.

Findings

Unbounded continua of solutions are analyzed between zeros of the nonlinearity.

Asymptotic behavior of solution continua with infinitely many zeros is characterized.

Certain nonlinearities exhibit that all positive parameters are bifurcation points, not necessarily branching points.

Abstract

In this paper, we study the behavior of multiple continua of solutions to the semilinear elliptic problem \begin{equation*} \begin{cases} -\Delta u = \lambda f(u) &\text{ in } \Omega, u=0 &\text{ on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded open subset of $\re^N$ and $f$ is a nonnegative continuous real function with multiple zeros. We analyze both the behavior of unbounded continua of solutions having norm between consecutive zeros of $f$, and the asymptotic behavior of the multiple unbounded continua in the case in which $f$ has a countable infinite set of positive zeros. In both cases, we pay special attention to the multiplicity results they give rise to. For the model cases $f(t) = t^r(1+\sin t)$ and $f(t) = t^r \left(1+\sin \frac{1}{t}\right)$ with $r\geq 0$ we show the surprising fact that there are some values of $r$ for which every $\lambda>0$ is a bifurcation point (either from infinity or from zero) that is not a branching point.