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

We show a sharp fractional Moser-Trudinger type inequality in dimension 1, i.e. for an interval $I \Subset \mathbb{R}, p \in (1,\infty)$ and some $\alpha > 0$
\[
\sup_{u\in\tilde{H}^{{1/p},p}(I):\|(-\Delta)^{1/2p} u\|_{L^p(I)} \le 1} \int_I |u|^a e^{\alpha_p |u|^{p/(p-1)} dx} < \infty \text{ if and only if } a = 0.
\]
Here $\tilde{H}^{{1/p},p}(I) = \{u \in L^p(\mathbb{R}) : (-\Delta)^{1/2p} u \in L^p(\mathbb{R}), \operatorname{supp}(u)\subset I\}$.
Restricting ourselves to the case p = 2 we further consider for $\lambda > 0$ the functional
\[
J(u) := 1/2 \int_{\mathbb{R}} |(-\Delta)^{1/4} u|^{2} dx - \lambda \int_I (e^{1/2 u^2} - 1) dx, u\in \tilde{H}^{{1/2},2}(I),
\]
and prove that it satisfies the Palais-Smale condition at any lever $c \in (-\infty, \pi)$. We use these results to show that the equation
\[
(-\Delta)^{1/2} u = \lambda u e^{1/2 u^2} \text{ in } I
\]
has a positive solution in $\tilde{H}^{{1/2},2}(I)$ if and only if $\lambda \in (0, \lambda_1 (I) )$, where $\lambda_1$ is the first eigenvalue of $(-\Delta)^{n/2}$ on I. This extends to the fractional case some previous results proven by Adimurthi for the Laplacian and the p-Laplacian operators.
Finally with a technique of Ruf we show a fractional Moser-Trudinger inequality on $\mathbb{R}$.