Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system

This article is concerned with the following Hamiltonian elliptic system: where is a small parameter, is a potential function, and is a super-quadratic sub-critical Hamiltonian. Applying suitable variational arguments and refined analysis techniques, we construct a new multiplicity result of semicla...

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Published inAdvances in nonlinear analysis Vol. 13; no. 1; pp. 97 - 100
Main Authors Zhang, Jian, Zhou, Huitao, Mi, Heilong
Format Journal Article
LanguageEnglish
Published De Gruyter 12.03.2024
Subjects
35J50
35Q40
58E05
Hamiltonian elliptic system
multiplicity
semiclassical solutions
Online AccessGet full text
ISSN2191-950X
2191-950X
DOI10.1515/anona-2023-0139

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Abstract This article is concerned with the following Hamiltonian elliptic system: where is a small parameter, is a potential function, and is a super-quadratic sub-critical Hamiltonian. Applying suitable variational arguments and refined analysis techniques, we construct a new multiplicity result of semiclassical solutions which depends on the number of global minimum points of . This result indicates how the shape of the graph of affects the number of semiclassical solutions.
AbstractList This article is concerned with the following Hamiltonian elliptic system: where is a small parameter, is a potential function, and is a super-quadratic sub-critical Hamiltonian. Applying suitable variational arguments and refined analysis techniques, we construct a new multiplicity result of semiclassical solutions which depends on the number of global minimum points of . This result indicates how the shape of the graph of affects the number of semiclassical solutions.
This article is concerned with the following Hamiltonian elliptic system: − ε 2 Δ u + ε b → ⋅ ∇ u + u + V ( x ) v = H v ( u , v ) in R N , − ε 2 Δ v − ε b → ⋅ ∇ v + v + V ( x ) u = H u ( u , v ) in R N , \left\{\begin{array}{l}-{\varepsilon }^{2}\Delta u+\varepsilon \overrightarrow{b}\cdot \nabla u+u+V\left(x)v={H}_{v}\left(u,v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ -{\varepsilon }^{2}\Delta v-\varepsilon \overrightarrow{b}\cdot \nabla v+v+V\left(x)u={H}_{u}\left(u,v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where ε > 0 \varepsilon \gt 0 is a small parameter, V V is a potential function, and H H is a super-quadratic sub-critical Hamiltonian. Applying suitable variational arguments and refined analysis techniques, we construct a new multiplicity result of semiclassical solutions which depends on the number of global minimum points of V V . This result indicates how the shape of the graph of V V affects the number of semiclassical solutions.
This article is concerned with the following Hamiltonian elliptic system: −ε2Δu+εb→⋅∇u+u+V(x)v=Hv(u,v)inRN,−ε2Δv−εb→⋅∇v+v+V(x)u=Hu(u,v)inRN,\left\{\begin{array}{l}-{\varepsilon }^{2}\Delta u+\varepsilon \overrightarrow{b}\cdot \nabla u+u+V\left(x)v={H}_{v}\left(u,v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ -{\varepsilon }^{2}\Delta v-\varepsilon \overrightarrow{b}\cdot \nabla v+v+V\left(x)u={H}_{u}\left(u,v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\end{array}\right. where ε>0\varepsilon \gt 0 is a small parameter, VV is a potential function, and HH is a super-quadratic sub-critical Hamiltonian. Applying suitable variational arguments and refined analysis techniques, we construct a new multiplicity result of semiclassical solutions which depends on the number of global minimum points of VV. This result indicates how the shape of the graph of VV affects the number of semiclassical solutions.
Author Zhang, Jian
Mi, Heilong
Zhou, Huitao
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  surname: Mi
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  organization: College of Science, Hunan University of Technology and Business, 410205 Changsha, Hunan, China
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Snippet This article is concerned with the following Hamiltonian elliptic system: where is a small parameter, is a potential function, and is a super-quadratic...
This article is concerned with the following Hamiltonian elliptic system: − ε 2 Δ u + ε b → ⋅ ∇ u + u + V ( x ) v = H v ( u , v ) in R N , − ε 2 Δ v − ε b → ⋅...
This article is concerned with the following Hamiltonian elliptic system:...
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StartPage 97
SubjectTerms 35J50
35Q40
58E05
Hamiltonian elliptic system
multiplicity
semiclassical solutions
Title Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
URI https://www.degruyter.com/doi/10.1515/anona-2023-0139
https://doaj.org/article/26dd223dad1149d7a917d301a865d151
Volume 13
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