Some existence and nonexistence results for a Schr\"odinger-Poisson type system

In this paper, we study the Schr\"odinger-Poisson system $$\left \{ \begin{array}{l} -\Delta u=\sqrt{p}u^{p-1}v, \quad u>0 \quad in \quad R^n, -\Delta v=\sqrt{p}u^p, \quad v>0 \quad in \quad R^n \end{array} \right.$$ with $n \geq 3$ and $p>1$. We investigate the existence and the nonexistence of positive classical solutions with the help of an integral system involving the Newton potential $$\left \{ \begin{array}{l} u(x)=c_1\displaystyle\int_{R^n}\frac{u^{p-1}(y)v(y)dy}{|x-y|^{n-2}}, \quad u>0 \quad in \quad R^n, v(x)=c_2\displaystyle\int_{R^n}\frac{u^p(y)dy}{|x-y|^{n-2}} \quad v>0 \quad in \quad R^n. \end{array} \right.$$ First, the system has no solution when $p\leq \frac{n}{n-2}$. When $p>\frac{n}{n-2}$, the system has a singular solution on $R^n \setminus \{0\}$ with slow asymptotic rate $\frac{2}{p-1}$. When $p<\frac{n+2}{n-2}$, the system has no solution in $L^{\frac{n(p-1)}{2}}(R^n)$. In fact, if the system has solutions in $L^{\frac{n(p-1)}{2}}(R^n)$, then $p=\frac{n+2}{n-2}$ and all the positive classical solutions can be classified as $u(x)=v(x)=c(\frac{t}{t^2+|x-x^*|^2})^{\frac{n-2}{2}}$, where $c,t$ are positive constants. When $p>\frac{n+2}{n-2}$, by the shooting method and the Pohozaev identity, we find another pair of radial solution $(u,v)$ satisfying $u \equiv v$ and decaying with slow rate $\frac{2}{p-1}$.