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Infinitely many bound state solutions of Schr\"{o}dinger-Poisson equations in $\mathbb{R}^3$
Keywords:Schr\"odinger-Poisson system, infinitely many solutions, without symmetric condition.
Abstract:
      In this paper, we study a system of Schr\"odinger-Poisson equation \[ \left\{ \begin{array}{c} -\Delta u+a(x)u+K(x)\phi u=|u|^{p-2}u,\quad \quad \quad \ \ \ \ \ \ x\in \mathbb{R}^3, \\ -\Delta \phi=K(x)u^2,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \ x\in \mathbb{R}^3, \end{array} \right. \] where $p\in (4,6)$ and $ K\geq (\not\equiv) 0$. Under some suitable decay assumptions but without any symmetry property on $a$ and $K$, we obtain infinitely many solutions of this system.