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 $ISubsetR{}$, $pin (1,infty)$ and some $alpha_p>0$ $$sup_{uin ilde H^{rac 1p,p}(I): |(-Delta)^rac{1}{2p}u|_{L^p(I)}le 1} int_I |u|^a e^{alpha_p |u|^rac{p}{p-1}}dx0$ the functional $$J(u):=rac{1}{2}int_{R{}}|(-Delta)^rac14 u|^2 dx- lambdaint_I left(e^{rac12 u^2}-1 ight)dx,quad uin ilde H^{rac{1}{2},2}(I),$$ and prove that it satisfies the Palais-Smale condition at any lever $cin (-infty,pi)$. We use these results to show that the equation $$(-Delta)^rac12 u =lambda u e^{rac{1}{2}u^2}quad ext{in }I$$ has a positive solution in $ ilde H^{rac12,2}(I)$ if and only if $lambdain (0,lambda_1(I))$, where $lambda_1(I)$ is the first eigenvalue of $(-Delta)^rac12$ 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 $R{}$.