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

Abstract

In this article, under some weaker assumptions on

and

f f

, 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

− Δ u + ϕ u = − a ∣ u ∣ p − 2 u + f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , \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 , 12 5 p\in \left(2,\frac{12}{5}\right)

. In particular, as a corollary for the following system:

− Δ u + ϕ u = − ∣ u ∣ p − 2 u + ∣ u ∣ q − 2 u , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , \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.