Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential

Abstract

In this article, under some weaker assumptions on a>0a\gt 0 and ff, the authors aim to study the existence of nontrivial radial solutions and nonexistence of nontrivial solutions for the following Schrödinger-Poisson system with zero mass potential −Δu+ϕu=−a∣u∣p−2u+f(u),x∈R3,−Δϕ=u2,x∈R3,\left\{\begin{array}{ll}-\Delta u+\phi u=-a{| u| }^{p-2}u+f\left(u),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. where p∈2,125p\in \left(2,\frac{12}{5}\right). In particular, as a corollary for the following system: −Δu+ϕu=−∣u∣p−2u+∣u∣q−2u,x∈R3,−Δϕ=u2,x∈R3,\left\{\begin{array}{ll}-\Delta u+\phi u=-{| u| }^{p-2}u+{| u| }^{q-2}u,& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. a sufficient and necessary condition is obtained on the existence of nontrivial radial solutions.

Keywords

• schrödinger-poisson system
• zero mass potential
• nonexistence
• 35j10
• 35j20