Synchronized vector solutions for the nonlinear Hartree system with nonlocal interaction

Abstract

We are concerned with the following nonlinear Hartree system−Δu+P1(|x|)u=α1|x|−1∗u2u+β|x|−1∗v2u inR3,−Δv+P2(|x|)v=α2|x|−1∗v2v+β|x|−1∗u2v inR3, $$\begin{cases}-{\Delta}u+{P}_{1}\left(\vert x\vert \right)u={\alpha }_{1}\left(\vert x{\vert }^{-1}\ast {u}^{2}\right)u+\beta \left(\vert x{\vert }^{-1}\ast {v}^{2}\right)u\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \\ -{\Delta}v+{P}_{2}\left(\vert x\vert \right)v={\alpha }_{2}\left(\vert x{\vert }^{-1}\ast {v}^{2}\right)v+\beta \left(\vert x{\vert }^{-1}\ast {u}^{2}\right)v\quad \hfill & \text{in} {\mathbb{R}}^{3},\hfill \end{cases}$$ where P 1(r) and P 2(r) are positive radial potentials, α 1 > 0, α 2 > 0 and β∈R $\beta \in \mathbb{R}$ is a coupling constant. We first study nondegeneracy of ground states for the limit system of the above problem. As applications, we show that the nonlinear Hartree system has infinitely many non-radial positive synchronized solutions, whose energy can be arbitrarily large.

Keywords

• synchronized vector solutions
• nonlinear hartree system
• reduction method
• primary 35j47, 35j60
• secondly 35b40, 35b45