We study the structure of solutions of the interior Bernoulli free boundary problem for \((-\Delta )^{\alpha /2}\) on an interval D with parameter \(\lambda > 0\). In particular, we show that there exists a constant \(\lambda _{\alpha ,D} > 0\) (called the Bernoulli constant) such that the problem has no solution for \(\lambda \in (0,\lambda _{\alpha ,D})\), at least one solution for \(\lambda = \lambda _{\alpha ,D}\) and at least two solutions for \(\lambda > \lambda _{\alpha ,D}\). We also study the interior Bernoulli problem for the fractional Laplacian for an interval with one free boundary point. We discuss the connection of the Bernoulli problem with the corresponding variational problem and present some conjectures. In particular, we show for \(\alpha = 1\) that there exists solutions of the interior Bernoulli free boundary problem for \((-\Delta )^{\alpha /2}\) on an interval which are not minimizers of the corresponding variational problem.

我们研究区间D上参数\(\lambda > 0\)的内部伯努利自由边界问题\((-\Delta )^{\alpha /2}\)的解的结构。特别地，我们证明存在一个常数\(\lambda _{\alpha ,D} > 0\)（称为伯努利常数），使得该问题对于\(\lambda \in (0,\lambda _{\alpha ,D})\) ， \(\lambda = \lambda _{\alpha ,D}\)的至少一个解和\(\lambda > \lambda _{\alpha的至少两个解， D}\)。我们还研究了具有一个自由边界点的区间的分数拉普拉斯的内部伯努利问题。我们讨论伯努利问题与相应变分问题的联系并提出一些猜想。特别是，我们证明对于\(\alpha = 1\) ，在区间上存在\((-\Delta )^{\alpha /2}\)的内部伯努利自由边界问题的解，该解不是相应的变分问题。