A fractional Moser-Trudinger type inequality in one dimension and its critical points. (Q265210)
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scientific article; zbMATH DE number 6562185
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A fractional Moser-Trudinger type inequality in one dimension and its critical points. |
scientific article; zbMATH DE number 6562185 |
Statements
A fractional Moser-Trudinger type inequality in one dimension and its critical points. (English)
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1 April 2016
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Moser-Trudinger-type inequality
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fractional derivative
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Sobolev space
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fractional Laplacian
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Bessel potential space
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0.96601933
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0.92317724
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0.9186008
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0.91804695
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0.9137091
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0.9137091
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0.9073403
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0.9052323
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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.
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