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

Let $b1 in (0,2)$, $x_0 in \mathbb{R}$, $r>0$, and $D=(x_0-r,x_0+r)$. 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.

Interval solution-count conjecture. 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 conjecture asserts a complete threshold description of the number of solutions for an interval. The preceding discussion indicates that it would follow from suitable monotonicity and convexity properties of the associated function $a8(a)$, but the required estimates, including numerical estimates, are beyond the scope of the paper.