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arXiv · 2412.11690

Intervals of bifurcation points for semilinear elliptic problems

José Carmona Tapia·Antonio J. Martínez Aparicio·Pedro J. Martínez-Aparicio
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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 $Ω$ 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 $λ>0$ is a bifurcation point (either from infinity or from zero) that is not a branching point.

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José Carmona Tapia, Antonio J. Martínez Aparicio, Pedro J. Martínez-Aparicio. 2024-12-16. Intervals of bifurcation points for semilinear elliptic problems. https://doi.org/10.1515/anona-2024-0061

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