Generalized Picone inequalities and their applications to (p,q)-Laplace equations
We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the (p,q)(p,q)-Laplace-type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide som...
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| Format: | Article |
| Language: | English |
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De Gruyter
2020-09-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2020-0065 |
| _version_ | 1818595979201347584 |
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| author | Bobkov Vladimir Tanaka Mieko |
| author_facet | Bobkov Vladimir Tanaka Mieko |
| author_sort | Bobkov Vladimir |
| collection | DOAJ |
| description | We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the (p,q)(p,q)-Laplace-type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation −Δpu−Δqu=fμ(x,u,∇u)-\hspace{-0.25em}{\text{Δ}}_{p}u-{\text{Δ}}_{q}u={f}_{\mu }(x,u,\nabla u) in a bounded domain Ω⊂ℝN\text{Ω}\hspace{0.25em}\subset {{\mathbb{R}}}^{N} under certain assumptions on the nonlinearity and with a special attention to the resonance case fμ(x,u,∇u)=λ1(p)|u|p−2u+μ|u|q−2u{f}_{\mu }(x,u,\nabla u)={\lambda }_{1}(p)|u{|}^{p-2}u+\mu |u{|}^{q-2}u, where λ1(p){\lambda }_{1}(p) is the first eigenvalue of the p-Laplacian. |
| first_indexed | 2024-12-16T11:24:37Z |
| format | Article |
| id | doaj.art-33d182fb97d145299b12939d626e9fb0 |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-12-16T11:24:37Z |
| publishDate | 2020-09-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-33d182fb97d145299b12939d626e9fb02022-12-21T22:33:23ZengDe GruyterOpen Mathematics2391-54552020-09-011811030104410.1515/math-2020-0065math-2020-0065Generalized Picone inequalities and their applications to (p,q)-Laplace equationsBobkov Vladimir0Tanaka Mieko1Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 301 00, Plzeň, Czech RepublicDepartment of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjyuku-ku, Tokyo, 162-8601, JapanWe obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the (p,q)(p,q)-Laplace-type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation −Δpu−Δqu=fμ(x,u,∇u)-\hspace{-0.25em}{\text{Δ}}_{p}u-{\text{Δ}}_{q}u={f}_{\mu }(x,u,\nabla u) in a bounded domain Ω⊂ℝN\text{Ω}\hspace{0.25em}\subset {{\mathbb{R}}}^{N} under certain assumptions on the nonlinearity and with a special attention to the resonance case fμ(x,u,∇u)=λ1(p)|u|p−2u+μ|u|q−2u{f}_{\mu }(x,u,\nabla u)={\lambda }_{1}(p)|u{|}^{p-2}u+\mu |u{|}^{q-2}u, where λ1(p){\lambda }_{1}(p) is the first eigenvalue of the p-Laplacian.https://doi.org/10.1515/math-2020-0065picone inequalitypicone identity(p,q)-laplaciannonexistencepositive solutions35j6235j2035p3035a01 |
| spellingShingle | Bobkov Vladimir Tanaka Mieko Generalized Picone inequalities and their applications to (p,q)-Laplace equations Open Mathematics picone inequality picone identity (p,q)-laplacian nonexistence positive solutions 35j62 35j20 35p30 35a01 |
| title | Generalized Picone inequalities and their applications to (p,q)-Laplace equations |
| title_full | Generalized Picone inequalities and their applications to (p,q)-Laplace equations |
| title_fullStr | Generalized Picone inequalities and their applications to (p,q)-Laplace equations |
| title_full_unstemmed | Generalized Picone inequalities and their applications to (p,q)-Laplace equations |
| title_short | Generalized Picone inequalities and their applications to (p,q)-Laplace equations |
| title_sort | generalized picone inequalities and their applications to p q laplace equations |
| topic | picone inequality picone identity (p,q)-laplacian nonexistence positive solutions 35j62 35j20 35p30 35a01 |
| url | https://doi.org/10.1515/math-2020-0065 |
| work_keys_str_mv | AT bobkovvladimir generalizedpiconeinequalitiesandtheirapplicationstopqlaplaceequations AT tanakamieko generalizedpiconeinequalitiesandtheirapplicationstopqlaplaceequations |