Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials in ℝ 3

Abstract

We study the following Schrödinger-Poisson system: $-\mathrm{\Delta }u+V(x)u+\varphi u=|u{|}^{p-\mathrm{1}}u$, $-\mathrm{\Delta }\varphi =u^{\mathrm{2}}$, ${\lim }_{|x|\to +\infty }\varphi (x)=\mathrm{0}$, where $u,\varphi :{ℝ}^{\mathrm{3}}\to ℝ$ are positive radial functions, $p\in (\mathrm{1},+\mathrm{\infty })$, $x=(x_{\mathrm{1}},x_{\mathrm{2}},x_{\mathrm{3}})\in {ℝ}^{\mathrm{3}}$, and $V(x)$ is allowed to take two different forms including $V(x)=\mathrm{1}/(x_{\mathrm{1}}^{\mathrm{2}}+x_{\mathrm{2}}^{\mathrm{2}}+x_{\mathrm{3}}^{\mathrm{2}}{)}^{\alpha /\mathrm{2}}$ and $V(x)=\mathrm{1}/(x_{\mathrm{1}}^{\mathrm{2}}+x_{\mathrm{2}}^{\mathrm{2}}{)}^{\alpha /\mathrm{2}}$ with $\alpha >\mathrm{0}$. Two theorems for nonexistence of nontrivial solutions are established, giving two regions on the $\alpha -p$ plane where the system has no nontrivial solutions.