Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight
Our main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi...
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MDPI AG
2024-03-01
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| author | Tao Zhang Jie Liu |
| author_facet | Tao Zhang Jie Liu |
| author_sort | Tao Zhang |
| collection | DOAJ |
| description | Our main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>β</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><msup><mrow><mo stretchy="false">[</mo><mo>−</mo><mo form="prefix">ln</mo><msup><mi>F</mi><mi>o</mi></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>β</mi></mrow></msup><mo>.</mo></mrow></semantics></math></inline-formula> This can be seen as a generation result of the isotropic Moser–Trudinger inequality with logarithmic weight. Furthermore, we obtain the existence of extremal function when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is small. Finally, we give Lions’ concentration-compactness principle, which is the improvement of the anisotropic Moser–Trudinger-type inequality. |
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| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-25T00:24:14Z |
| publishDate | 2024-03-01 |
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| spelling | doaj.art-4c3d8b33836b46b491a8d5893c0c9f322024-03-12T16:50:22ZengMDPI AGMathematics2227-73902024-03-0112578510.3390/math12050785Anisotropic Moser–Trudinger-Type Inequality with Logarithmic WeightTao Zhang0Jie Liu1School of Mathematics and Information Science, Yantai University, Yantai 264005, ChinaSchool of Mathematics and Information Science, Yantai University, Yantai 264005, ChinaOur main purpose in this paper is to study the anisotropic Moser–Trudinger-type inequalities with logarithmic weight <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mi>β</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><msup><mrow><mo stretchy="false">[</mo><mo>−</mo><mo form="prefix">ln</mo><msup><mi>F</mi><mi>o</mi></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>β</mi></mrow></msup><mo>.</mo></mrow></semantics></math></inline-formula> This can be seen as a generation result of the isotropic Moser–Trudinger inequality with logarithmic weight. Furthermore, we obtain the existence of extremal function when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is small. Finally, we give Lions’ concentration-compactness principle, which is the improvement of the anisotropic Moser–Trudinger-type inequality.https://www.mdpi.com/2227-7390/12/5/785anisotropic Moser–Trudinger-type inequalitylogarithmic weightexistence of extremal function |
| spellingShingle | Tao Zhang Jie Liu Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight Mathematics anisotropic Moser–Trudinger-type inequality logarithmic weight existence of extremal function |
| title | Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight |
| title_full | Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight |
| title_fullStr | Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight |
| title_full_unstemmed | Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight |
| title_short | Anisotropic Moser–Trudinger-Type Inequality with Logarithmic Weight |
| title_sort | anisotropic moser trudinger type inequality with logarithmic weight |
| topic | anisotropic Moser–Trudinger-type inequality logarithmic weight existence of extremal function |
| url | https://www.mdpi.com/2227-7390/12/5/785 |
| work_keys_str_mv | AT taozhang anisotropicmosertrudingertypeinequalitywithlogarithmicweight AT jieliu anisotropicmosertrudingertypeinequalitywithlogarithmicweight |