Ground state solutions for Hamiltonian elliptic system with inverse square potential

By Jian Zhang, Wen Zhang, Xianhua Tang. 2017

84 citations recorded by the source corpus.

In this paper, we study the following Hamiltonian elliptic system with gradient term and inverse square potential \begin{document}$ \left\{ \begin{array}{ll}-\Delta u +\vec{b}(x)\cdot \nabla u +V(x)u-\frac{\mu}{|x|^{2}}v=H_{v}(x,u,v)\\-\Delta v -\vec{b}(x)\cdot \nabla v +V(x)v-\frac{\mu}{|x|^{2}}u=H_{u}(x,u,v)\\\end{array} \right.$\end{document} for $x\in\mathbb{R}^{N}$, where $N\geq3$, $\mu\in\mathbb{R}$, and $V(x)$, $\vec{b}(x)$ and $H(x, u, v)$ are $1$-periodic in $x$. Under suitable conditions, we prove that the system possesses a ground state solution via variational methods for sufficiently small $\mu\geq0$. Moreover, we provide the comparison of the energy of ground state solutions for the case $\mu>0$ and $\mu=0$. Finally, we also give the convergence property of ground state solutions as $\mu\to0^+$.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5e6cd8273e8741cc9d8a12d00e2f8a7956df266b