20 research outputs found
Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension
Given a smoothly bounded domain with
odd, we study the blow-up of bounded sequences of solutions to the non-local equation
where , and denotes the Lions-Magenes spaces of functions which are supported in and with
. Extending previous works of
Druet, Robert-Struwe and the second author, we show that if the sequence
is not bounded in , a suitably rescaled subsequence
converges to the function
, which solves the prescribed
non-local -curvature equation recently studied by Da
Lio-Martinazzi-Rivi\`ere when , Jin-Maalaoui-Martinazzi-Xiong when ,
and Hyder when is odd. We infer that blow-up can occur only if
\epsilon-regularity for systems involving non-local, antisymmetric operators
We prove an epsilon-regularity theorem for critical and super-critical
systems with a non-local antisymmetric operator on the right-hand side.
These systems contain as special cases, Euler-Lagrange equations of
conformally invariant variational functionals as Rivi\`ere treated them, and
also Euler-Lagrange equations of fractional harmonic maps introduced by Da
Lio-Rivi\`ere.
In particular, the arguments presented here give new and uniform proofs of
the regularity results by Rivi\`ere, Rivi\`ere-Struwe, Da-Lio-Rivi\`ere, and
also the integrability results by Sharp-Topping and Sharp, not discriminating
between the classical local, and the non-local situations