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We construct solutions for the fractional Yamabe problem in the$n$-dimensional Euclidean space which are singular along a smooth submanifoldof dimension $(n-2s)/2$ and lead to complete metrics. Here $s\in(0,1)$ is thefractional exponent and we restrict to $s=1/2$ for parity reasons. We perform adirect integral asymptotic analysis with global geometric weights, based onstability and semilinearity of the equation. This completes the missing case ina previous work we did in collaboration with Ao, Fontelos, Gonz\'{a}lez andWei, where complete singular metrics of dimensions $0<k<(n-2s)/2$ wereconstructed. Moreover the dimension $(n-2s)/2$ is maximal for the conformalfactor to be a distributional solution. Even so, we conjecture that there existcomplete metrics with prescribed singularities of strictly higher dimensions.We also provide a complete study of the model problem -- the fractionalLane--Emden equation posed in the punctured ball with Serrin's criticalexponent and homogeneous Dirichlet condition, for all $s\in(0,1)$. First, weconstruct the profile, show that it is radially symmetric, and build multiplesolutions in a bounded domain with any prescribed closed singular set.Secondly, we prove a Liouville-type result, asserting the non-existence ofexterior solutions of the model equation without any assumption on itsasymptotic behavior. Finally, we develop new methods to prove that the singularbehavior of the profile is unique. In sharp contrast to known techniques in theliterature, our method is based on the striking connection between thenon-local equation and its associated first order ODE. A local continuationargument is then utilized. This gives an alternative proof to the recent workof Wei and Wu based on adapting the classical proof in the extension, and itseems to be the first time that a genuinely local ODE is used to study anon-local elliptic problem.

From fractional Lane-Emden-Serrin equation -- existence, multiplicity  and local behaviors via classical ODE -- to fractional Yamabe metrics with  singularity of "maximal" dimension