Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities
Abstract We investigate the multiplicity of solutions for one-dimensional p-Laplacian Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems: The first states that there exists exactly one solution when nonlinearities cross no eigenvalue. The second guarantees that th...
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Online Access: | http://link.springer.com/article/10.1186/s13661-019-1165-5 |
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doaj-793f013a135a4a21ab35f38870ff9a5a2020-11-25T02:15:36ZengSpringerOpenBoundary Value Problems1687-27702019-03-012019111410.1186/s13661-019-1165-5Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearitiesTacksun Jung0Q-Heung Choi1Department of Mathematics, Kunsan National UniversityDepartment of Mathematics Education, Inha UniversityAbstract We investigate the multiplicity of solutions for one-dimensional p-Laplacian Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems: The first states that there exists exactly one solution when nonlinearities cross no eigenvalue. The second guarantees that there exist exactly two solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross just the first eigenvalue. The third claims that there exist at least three solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross the first and second eigenvalues. We obtain the first and second theorem by considering the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem, and the contraction mapping principle in the p-Lebesgue space (when p≥2 $p\ge 2$). We obtain the third result by Leray–Schauder degree theory.http://link.springer.com/article/10.1186/s13661-019-1165-5p-Laplacian problemp-Laplacian eigenvalue problemJumping nonlinearityContraction mapping principleLeray–Schauder degree theory |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Tacksun Jung Q-Heung Choi |
spellingShingle |
Tacksun Jung Q-Heung Choi Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities Boundary Value Problems p-Laplacian problem p-Laplacian eigenvalue problem Jumping nonlinearity Contraction mapping principle Leray–Schauder degree theory |
author_facet |
Tacksun Jung Q-Heung Choi |
author_sort |
Tacksun Jung |
title |
Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities |
title_short |
Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities |
title_full |
Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities |
title_fullStr |
Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities |
title_full_unstemmed |
Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities |
title_sort |
multiplicity results for p-laplacian boundary value problem with jumping nonlinearities |
publisher |
SpringerOpen |
series |
Boundary Value Problems |
issn |
1687-2770 |
publishDate |
2019-03-01 |
description |
Abstract We investigate the multiplicity of solutions for one-dimensional p-Laplacian Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems: The first states that there exists exactly one solution when nonlinearities cross no eigenvalue. The second guarantees that there exist exactly two solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross just the first eigenvalue. The third claims that there exist at least three solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross the first and second eigenvalues. We obtain the first and second theorem by considering the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem, and the contraction mapping principle in the p-Lebesgue space (when p≥2 $p\ge 2$). We obtain the third result by Leray–Schauder degree theory. |
topic |
p-Laplacian problem p-Laplacian eigenvalue problem Jumping nonlinearity Contraction mapping principle Leray–Schauder degree theory |
url |
http://link.springer.com/article/10.1186/s13661-019-1165-5 |
work_keys_str_mv |
AT tacksunjung multiplicityresultsforplaplacianboundaryvalueproblemwithjumpingnonlinearities AT qheungchoi multiplicityresultsforplaplacianboundaryvalueproblemwithjumpingnonlinearities |
_version_ |
1724895119550709760 |