Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems
In this thesis, we research quasilinear Schrödinger system as follows in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo><</mo><mi>N</mi><mo>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-03-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/13/3/182 |
| _version_ | 1797242063387885568 |
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| author | Xue Zhang Jing Zhang |
| author_facet | Xue Zhang Jing Zhang |
| author_sort | Xue Zhang |
| collection | DOAJ |
| description | In this thesis, we research quasilinear Schrödinger system as follows in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo><</mo><mi>N</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo><</mo><mi>q</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mn>1</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><msub><mi>V</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> are continuous functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>,</mo><mi>ι</mi></mrow></semantics></math></inline-formula> are parameters with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>,</mo><mi>ι</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and nonlinear terms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>×</mo><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup><mo>,</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. We find a nontrivial solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo>></mo><msub><mi>ι</mi><mn>1</mn></msub><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> by means of the mountain-pass theorem and change of variable theorem. Our main novelty of the thesis is that we extend <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Δ</mo><mi>p</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Δ</mo><mi>q</mi></msub></semantics></math></inline-formula> to find the existence of a nontrivial solution. |
| first_indexed | 2024-04-24T18:33:16Z |
| format | Article |
| id | doaj.art-468656010eb040218cd3955ed4ec486c |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-04-24T18:33:16Z |
| publishDate | 2024-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-468656010eb040218cd3955ed4ec486c2024-03-27T13:21:02ZengMDPI AGAxioms2075-16802024-03-0113318210.3390/axioms13030182Nontrivial Solutions for a Class of Quasilinear Schrödinger SystemsXue Zhang0Jing Zhang1College of Mathematics Science, Inner Mongolia Normal University, Hohhot 010011, ChinaCollege of Mathematics Science, Inner Mongolia Normal University, Hohhot 010011, ChinaIn this thesis, we research quasilinear Schrödinger system as follows in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>3</mn><mo><</mo><mi>N</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo><</mo><mi>q</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>V</mi><mn>1</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><msub><mi>V</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> are continuous functions, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>,</mo><mi>ι</mi></mrow></semantics></math></inline-formula> are parameters with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>,</mo><mi>ι</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, and nonlinear terms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>,</mo><mi>h</mi><mo>∈</mo><mi>C</mi><mo stretchy="false">(</mo><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mo>×</mo><msup><mi mathvariant="double-struck">R</mi><mn>2</mn></msup><mo>,</mo><mi mathvariant="double-struck">R</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. We find a nontrivial solution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo>></mo><msub><mi>ι</mi><mn>1</mn></msub><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> by means of the mountain-pass theorem and change of variable theorem. Our main novelty of the thesis is that we extend <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Δ</mo></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Δ</mo><mi>p</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Δ</mo><mi>q</mi></msub></semantics></math></inline-formula> to find the existence of a nontrivial solution.https://www.mdpi.com/2075-1680/13/3/182change of variablenontrivial solutionmountain-pass theorem |
| spellingShingle | Xue Zhang Jing Zhang Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems Axioms change of variable nontrivial solution mountain-pass theorem |
| title | Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems |
| title_full | Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems |
| title_fullStr | Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems |
| title_full_unstemmed | Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems |
| title_short | Nontrivial Solutions for a Class of Quasilinear Schrödinger Systems |
| title_sort | nontrivial solutions for a class of quasilinear schrodinger systems |
| topic | change of variable nontrivial solution mountain-pass theorem |
| url | https://www.mdpi.com/2075-1680/13/3/182 |
| work_keys_str_mv | AT xuezhang nontrivialsolutionsforaclassofquasilinearschrodingersystems AT jingzhang nontrivialsolutionsforaclassofquasilinearschrodingersystems |