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

Abstract

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.

Keywords

• hamiltonian elliptic system
• semiclassical solutions
• multiplicity
• 35j50
• 35q40
• 58e05