Intervals of bifurcation points for semilinear elliptic problems
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arXiv
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| Format: | Preprint |
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2024
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| author | Tapia, José Carmona Aparicio, Antonio J. Martínez Martínez-Aparicio, Pedro J. |
| author_facet | Tapia, José Carmona Aparicio, Antonio J. Martínez Martínez-Aparicio, Pedro J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11690 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intervals of bifurcation points for semilinear elliptic problems Tapia, José Carmona Aparicio, Antonio J. Martínez Martínez-Aparicio, Pedro J. Analysis of PDEs 35B32, 35B40, 35J25, 35J61 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. |
| title | Intervals of bifurcation points for semilinear elliptic problems |
| topic | Analysis of PDEs 35B32, 35B40, 35J25, 35J61 |
| url | https://arxiv.org/abs/2412.11690 |