A fractional Moser-Trudinger type inequalitiy in one dimension and its critical points

We show a sharp fractional Moser-Trudinger type inequality in dimension 1, i.e., for any interval I⋐R and p∈(1,∞) there exists αp>0 such that
supu∈H~1p,p(I):∥(−Δ)12pu∥Lp(I)≤1∫Ieαp|u|pp−1dx=Cp|I|,
and αp is optimal in the sense that
supu∈H~1p,p(I):∥(−Δ)12pu∥Lp(I)≤1∫Ih(u)eαp|u|pp−1dx=+∞,
for any function h:[0,∞)→[0,∞) with limt→∞h(t)=∞. Here, H~1p,p(I)={u∈Lp(R):(−Δ)12pu∈Lp(R),supp(u)⊂I¯}. Restricting ourselves to the case p=2, we further consider for λ>0 the functional
J(u):=12∫R|(−Δ)14u|2dx−λ∫I(e12u2−1)dx,u∈H~12,2(I),
and prove that it satisfies the Palais-Smale condition at any level c∈(−∞,π). We use these results to show that the equation
(−Δ)12u=λue12u2in I,
has a positive solution in H~12,2(I) if and only if λ∈(0,λ1(I)), where λ1(I) is the first eigenvalue of (−Δ)12 on I. This extends to the fractional case for some previous results proven by Adimurthi for the Laplacian and the p-Laplacian operators. Finally, with a technique by Ruf, we show a fractional Moser-Trudinger inequality on R.