On sign-changing solutions for (p,q)-Laplace equations with two parameters
We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous (p,q){(p,q)}-Laplace equations -Δpu-Δqu=α|u|p-2u+β|u|q-2u{-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2% }u} where p≠q{p...
Main Authors: | , |
---|---|
Format: | Article |
Language: | English |
Published: |
De Gruyter
2016-12-01
|
Series: | Advances in Nonlinear Analysis |
Subjects: | |
Online Access: | https://doi.org/10.1515/anona-2016-0172 |
_version_ | 1818726626183086080 |
---|---|
author | Bobkov Vladimir Tanaka Mieko |
author_facet | Bobkov Vladimir Tanaka Mieko |
author_sort | Bobkov Vladimir |
collection | DOAJ |
description | We investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous (p,q){(p,q)}-Laplace equations -Δpu-Δqu=α|u|p-2u+β|u|q-2u{-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2%
}u} where p≠q{p\neq q}. By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the (α,β){(\alpha,\beta)}-plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p- and q-Laplacians in one dimension. |
first_indexed | 2024-12-17T22:01:11Z |
format | Article |
id | doaj.art-40d99fe860bd4fde89ce6599792d89cc |
institution | Directory Open Access Journal |
issn | 2191-9496 2191-950X |
language | English |
last_indexed | 2024-12-17T22:01:11Z |
publishDate | 2016-12-01 |
publisher | De Gruyter |
record_format | Article |
series | Advances in Nonlinear Analysis |
spelling | doaj.art-40d99fe860bd4fde89ce6599792d89cc2022-12-21T21:30:59ZengDe GruyterAdvances in Nonlinear Analysis2191-94962191-950X2016-12-018110112910.1515/anona-2016-0172anona-2016-0172On sign-changing solutions for (p,q)-Laplace equations with two parametersBobkov Vladimir0Tanaka Mieko1Institute of Mathematics, Ufa Scientific Center, Russian Academy of Sciences, Chernyshevsky str. 112, Ufa450008, Russia; and Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, Plzeň 306 14, Czech RepublicDepartment of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjyuku-ku, Tokyo162-8601, JapanWe investigate the existence of nodal (sign-changing) solutions to the Dirichlet problem for a two-parametric family of partially homogeneous (p,q){(p,q)}-Laplace equations -Δpu-Δqu=α|u|p-2u+β|u|q-2u{-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2% }u} where p≠q{p\neq q}. By virtue of the Nehari manifolds, the linking theorem, and descending flow, we explicitly characterize subsets of the (α,β){(\alpha,\beta)}-plane which correspond to the existence of nodal solutions. In each subset the obtained solutions have prescribed signs of energy and, in some cases, exactly two nodal domains. The nonexistence of nodal solutions is also studied. Additionally, we explore several relations between eigenvalues and eigenfunctions of the p- and q-Laplacians in one dimension.https://doi.org/10.1515/anona-2016-0172eigenvalue problemfirst eigenvaluesecond eigenvaluenodal solutionssign-changing solutionsnehari manifoldlinking theoremdescending flow35j62 35j20 35p30 |
spellingShingle | Bobkov Vladimir Tanaka Mieko On sign-changing solutions for (p,q)-Laplace equations with two parameters Advances in Nonlinear Analysis eigenvalue problem first eigenvalue second eigenvalue nodal solutions sign-changing solutions nehari manifold linking theorem descending flow 35j62 35j20 35p30 |
title | On sign-changing solutions for (p,q)-Laplace equations with two parameters |
title_full | On sign-changing solutions for (p,q)-Laplace equations with two parameters |
title_fullStr | On sign-changing solutions for (p,q)-Laplace equations with two parameters |
title_full_unstemmed | On sign-changing solutions for (p,q)-Laplace equations with two parameters |
title_short | On sign-changing solutions for (p,q)-Laplace equations with two parameters |
title_sort | on sign changing solutions for p q laplace equations with two parameters |
topic | eigenvalue problem first eigenvalue second eigenvalue nodal solutions sign-changing solutions nehari manifold linking theorem descending flow 35j62 35j20 35p30 |
url | https://doi.org/10.1515/anona-2016-0172 |
work_keys_str_mv | AT bobkovvladimir onsignchangingsolutionsforpqlaplaceequationswithtwoparameters AT tanakamieko onsignchangingsolutionsforpqlaplaceequationswithtwoparameters |