Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

TL;DR

This paper demonstrates the existence of nontrivial, unbounded, overdetermined elliptic problems in certain domains bifurcating from cylinders, for a broad class of nonlinearities, using local bifurcation techniques.

Contribution

It establishes the existence of solutions in complex domains for general nonlinearities, extending previous results to more general settings.

Findings

Existence of solutions in unbounded domains bifurcating from cylinders.

Applicability to a broad class of nonlinear functions.

Use of local bifurcation methods to prove results.

Abstract

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the overdetermined elliptic problem \begin{equation*} \begin{cases} \Delta u +f(u)=0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=\mbox{constant} &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} has a positive bounded solution. We will prove such result for a very general class of functions $f: [0, +\infty) \to \mathbb{R}$. Roughly speaking, we only ask that the Dirichlet problem in $B$ admits a nondegenerate solution. The proof uses a local bifurcation argument.