A fractional Moser-Trudinger type inequality in one dimension and its critical points.
The authors investigate fractional Moser-Trudinger-type inequalities on intervals \(I\Subset \mathbb{R}\) and show that the inequality is optimal in a certain sense. Furthermore, they establish that the equation NEWLINE\[NEWLINE(-\Delta)^\frac12 u = \lambda u \exp (\tfrac12 u^2)\qquad\text{in \(I\Subset \mathbb{R}\)},NEWLINE\]NEWLINE has a positive solution in the Bessel potential space \(\widetilde{H}^{\frac12,2} (I)\) iff \(\lambda\in (0, \lambda_1(I))\), where \(\lambda_1(I)\) denotes the first eigenvalue of \((-\Delta)^\frac12\) in \(I\), extending a previous result by \textit{Adimurthi} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No. 3, 393-413 (1990; Zbl 0732.35028)]. Finally, a fractional Moser-Trudinger type inequality on \(\mathbb R\) is derived.
scientific article; zbMATH DE number 6562185