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arXiv · 2304.04525

Interval bifurcation theorems for Fredholm operator and its application to an elliptic overdetermined problem in bounded domains

Guowei Dai·Yong Zhang
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Abstract

We establish local interval bifurcation theorem and global interval bifurcation theorem for Fredholm operator with index $0$ via $0$-group. As one of applications, we investigate the existence of a family of nontrivial domains $\Omega_{\rho}\subset \mathbb{R}^N$ ($N=2,3$ or $4$), bifurcating from a small ball, such that the problem \begin{equation} -\Delta u=u-\left(u^+\right)^3\,\, \text{in}\,\,\Omega_{\rho}, \,\, u=0,\,\,\partial_\nu u=\text{const}\,\,\text{on}\,\,\partial\Omega_{\rho} \nonumber \end{equation} has a sign-changing bounded solution. Compared with the recent result \cite[Theorem 2.1]{Ruiz}, here we obtain a family of domains $\Omega_{\rho}$ instead of a sequence of domains.

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Guowei Dai, Yong Zhang. 2023-04-10. Interval bifurcation theorems for Fredholm operator and its application to an elliptic overdetermined problem in bounded domains. https://arxiv.org/abs/2304.04525

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