Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
2024; De Gruyter; Volume: 13; Issue: 1 Linguagem: Inglês
10.1515/anona-2023-0139
ISSN2191-950X
AutoresJian Zhang, Huitao Zhou, Heilong Mi,
Tópico(s)Advanced Mathematical Modeling in Engineering
ResumoAbstract 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.
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