On the non-autonomous Schr\"odinger-Poisson problems in $\mathbb{R}^{3}$

Abstract

In this paper, we study the problem: \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+u+\lambda K\left( x\right) \phi u=a\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =K\left( x\right) u^{2} & \ \text{in }\mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\lambda >0$ and $2<p<4$. We require that $K\left( x\right)$ and $a\left( x\right) $ are nonnegative functions in $\mathbb{R}^{3}$ and satisfy some suitable assumptions, but not requiring any symmetry property on them. Assuming that $\lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) =K_{\infty }\geq 0$ and $\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) =a_{\infty }>0$, we establish some existence results of positive solutions, depending on the parameter $\lambda$. More importantly, we prove the existence of ground state solutions for the case $3.18\thickapprox \frac{1+\sqrt{73}}{3}<p<4.$