<p>We study boundedness, optimality and attainability of Trudinger-Moser type maximization problems in the radial and the subcritical homogeneous Sobolev spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1062_Article_IEq1.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="143" /> </InlineMediaObject> <EquationSource Format="TEX">${\dot{W}}^{1,p}_{0, \text {rad}}(B_R^N)\,(p<N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mover accent="true"> <mi>W</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mn>0</mn> <mo>,</mo> <mtext>rad</mtext> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>B</mi> <mi>R</mi> <mi>N</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo><</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Our results give a revision of an error in [<CitationRef CitationID="CR13">13</CitationRef>, Theorem C]. Also, our inequality converges to the original Trudinger-Moser inequality as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="30_2025_1062_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$p \nearrow N$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>↗</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> including optimal exponent and concentration limit. Finally, we consider an application of our inequality to elliptic problems with exponential nonlinearity.</p>

Weighted Trudinger-Moser inequalities in the subcritical Sobolev spaces and their applications

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2025年6月23日

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Masahiro Ikeda,
Megumi Sano,
Koichi Taniguchi

摘要

We study boundedness, optimality and attainability of Trudinger-Moser type maximization problems in the radial and the subcritical homogeneous Sobolev spaces ${\dot{W}}^{1,p}_{0, \text {rad}}(B_R^N)\,(p<N)$ ${\dot{W}}_{0, rad}^{1, p} (B_{R}^{N}) (p < N)$ . Our results give a revision of an error in [13, Theorem C]. Also, our inequality converges to the original Trudinger-Moser inequality as $p \nearrow N$ $p ↗ N$ including optimal exponent and concentration limit. Finally, we consider an application of our inequality to elliptic problems with exponential nonlinearity.