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 -Δp⁢u-Δq⁢u=α⁢|u|p-2⁢u+β⁢|u|q-2⁢u{-\Delta_{p}u-\Delta_{q}u=\alpha\lvert u\rvert^{p-2}u+\beta\lvert u\rvert^{q-2% }u} where p≠q{p...

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Main Authors: Bobkov Vladimir, Tanaka Mieko
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
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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 -Δp⁢u-Δq⁢u=α⁢|u|p-2⁢u+β⁢|u|q-2⁢u{-\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.
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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 -Δp⁢u-Δq⁢u=α⁢|u|p-2⁢u+β⁢|u|q-2⁢u{-\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