Intervals of bifurcation points for semilinear elliptic problems

Abstract

In this paper, we study the behavior of multiple continua of solutions to the semilinear elliptic problem \begin{equation*} \begin{cases} -Δu = λf(u) &\text{ in } Ω, u=0 &\text{ on } \partialΩ, \end{cases} \end{equation*} where $\hat{I}\text{c◯}$ 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 $r\ge 0$ we show the surprising fact that there are some values of $r$ for which every $\hat{I}»>0$ is a bifurcation point (either from infinity or from zero) that is not a branching point.