Nonlinear Eigenvalue Problems for the Dirichlet (<i>p</i>,2)-Laplacian
We consider a nonlinear eigenvalue problem driven by the Dirichlet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mn>2</mn>&l...
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2022-01-01
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| Online Access: | https://www.mdpi.com/2075-1680/11/2/58 |
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| author | Yunru Bai Leszek Gasiński Nikolaos S. Papageorgiou |
| author_facet | Yunru Bai Leszek Gasiński Nikolaos S. Papageorgiou |
| author_sort | Yunru Bai |
| collection | DOAJ |
| description | We consider a nonlinear eigenvalue problem driven by the Dirichlet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>-Laplacian. The parametric reaction is a Carathéodory function which exhibits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-sublinear growth as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></semantics></math></inline-formula>. Using variational tools and truncation and comparison techniques, we prove a bifurcation-type theorem describing the “spectrum” as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> varies. We also prove the existence of a smallest positive eigenfunction for every eigenvalue. Finally, we indicate how the result can be extended to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-equations (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≠</mo><mn>2</mn></mrow></semantics></math></inline-formula>). |
| first_indexed | 2024-03-09T22:36:29Z |
| format | Article |
| id | doaj.art-2765b770a8ef48eab0b233c5eb0b6e6e |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T22:36:29Z |
| publishDate | 2022-01-01 |
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| spelling | doaj.art-2765b770a8ef48eab0b233c5eb0b6e6e2023-11-23T18:47:00ZengMDPI AGAxioms2075-16802022-01-011125810.3390/axioms11020058Nonlinear Eigenvalue Problems for the Dirichlet (<i>p</i>,2)-LaplacianYunru Bai0Leszek Gasiński1Nikolaos S. Papageorgiou2School of Science, Guangxi University of Science and Technology, Liuzhou 545006, ChinaDepartment of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30-084 Cracow, PolandDepartment of Mathematics, National Technical University, Zografou Campus, 15780 Athens, GreeceWe consider a nonlinear eigenvalue problem driven by the Dirichlet <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>-Laplacian. The parametric reaction is a Carathéodory function which exhibits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-sublinear growth as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>→</mo><mo>+</mo><mo>∞</mo></mrow></semantics></math></inline-formula> and as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></semantics></math></inline-formula>. Using variational tools and truncation and comparison techniques, we prove a bifurcation-type theorem describing the “spectrum” as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> varies. We also prove the existence of a smallest positive eigenfunction for every eigenvalue. Finally, we indicate how the result can be extended to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-equations (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>≠</mo><mn>2</mn></mrow></semantics></math></inline-formula>).https://www.mdpi.com/2075-1680/11/2/58(<i>p</i>,2) and (<i>p</i>,<i>q</i>)-Laplaciansnonlinear regularitypositive solutionsstrong comparison principlesublinear reactionbifurcation-type results |
| spellingShingle | Yunru Bai Leszek Gasiński Nikolaos S. Papageorgiou Nonlinear Eigenvalue Problems for the Dirichlet (<i>p</i>,2)-Laplacian Axioms (<i>p</i>,2) and (<i>p</i>,<i>q</i>)-Laplacians nonlinear regularity positive solutions strong comparison principle sublinear reaction bifurcation-type results |
| title | Nonlinear Eigenvalue Problems for the Dirichlet (<i>p</i>,2)-Laplacian |
| title_full | Nonlinear Eigenvalue Problems for the Dirichlet (<i>p</i>,2)-Laplacian |
| title_fullStr | Nonlinear Eigenvalue Problems for the Dirichlet (<i>p</i>,2)-Laplacian |
| title_full_unstemmed | Nonlinear Eigenvalue Problems for the Dirichlet (<i>p</i>,2)-Laplacian |
| title_short | Nonlinear Eigenvalue Problems for the Dirichlet (<i>p</i>,2)-Laplacian |
| title_sort | nonlinear eigenvalue problems for the dirichlet i p i 2 laplacian |
| topic | (<i>p</i>,2) and (<i>p</i>,<i>q</i>)-Laplacians nonlinear regularity positive solutions strong comparison principle sublinear reaction bifurcation-type results |
| url | https://www.mdpi.com/2075-1680/11/2/58 |
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