Some existence and nonexistence results for a Schrödinger-Poisson type system

Abstract

In this paper, we study the Schrödinger-Poisson system $$\{\begin{array}{l}-\Delta u=\sqrt{p}{u}^{p-1}v,u>0in{R}^n,-\Delta v=\sqrt{p}{u}^p,v>0in{R}^n\end{array}$$ with $n\ge 3$ 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 $$\{\begin{array}{l}u(x)=c_1{\int }_{R^n}\frac{u^{p-1}(y)v(y)dy}{|x-y{|}^{n-2}},u>0in{R}^n,v(x)=c_2{\int }_{R^n}\frac{u^p(y)dy}{|x-y{|}^{n-2}}v>0in{R}^n.\end{array}$$ First, the system has no solution when $p\le \frac{n}{n-2}$. 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 $u(x)=v(x)=c(\frac{t}{t^2+|x-x^{\ast }{|}^2}{)}^{\frac{n-2}{2}}$, 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 $\frac{2}{p-1}$.