The ball inner Bernoulli solution-count conjecture for the fractional Laplacian

Let $b1 i (0,2)$, $d a 2$, $x_0 i\tilde{\mathbb{R}}^d$, $r>0$, and let $D={x i\tilde{\mathbb{R}}^d:|x-x_0|<r}$. The inner Bernoulli problem for the fractional Laplacian on $D$, with parameter $bbda$, is the problem referred to as Problem ref{bernoulli_problem} in the source.

Ball solution-count hypothesis. There exists a constant $bbda_{b1,D}>0$ such that the problem has exactly two solutions for $bbda>bbda_{b1,D}$, exactly one solution for $bbda=bbda_{b1,D}$, and no solution for $bbda<bbda_{b1,D}$.

The hypothesis is motivated by known results for the classical inner Bernoulli problem for balls. Its validity for the fractional Laplacian is presented as an unproved extension to higher-dimensional balls.