Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
Jian Zhang, Huitao Zhou, Heilong Mi
Abstract
Abstract This article is concerned with the following Hamiltonian elliptic system: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:msup> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>ε</m:mi> <m:mover accent="true"> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mo>→</m:mo> </m:mrow> </m:mover> <m:mo>⋅</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>v</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mi>v</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mspace width="1em"/> <m:mspace width="0.1em"/> <m:mtext>in</m:mtext> <m:mspace width="0.1em"/> <m:mspace width="0.33em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:msup> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>v</m:mi> <m:mo>−</m:mo> <m:mi>ε</m:mi> <m:mover accent="true"> <m:mrow> <m:mi>b</m:mi> </m:mrow> <m:mrow> <m:mo>→</m:mo> </m:mrow> </m:mover> <m:mo>⋅</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>v</m:mi> <m:mo>+</m:mo> <m:mi>v</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mspace width="1em"/> <m:mspace width="0.1em"/> <m:mtext>in</m:mtext> <m:mspace width="0.1em"/> <m:mspace width="0.33em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> \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 <jats:inline-graphic xmlns:xlink="http://www.w3.or