Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent

Abstract

This article focuses on the study of the following Schrödinger-Poisson system with zero mass: −Δu+ϕu=∣u∣u+f(u),x∈R3,−Δϕ=u2,x∈R3,\left\{\begin{array}{ll}-\Delta u+\phi u=| u| u+f\left(u),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. where ff is a continuous function satisfying some general growth conditions, and it requires only to be super-quadratic growth at infinity and includes, in particular, the pure power function ∣u∣p−2u{| u| }^{p-2}u with p∈(3,6)p\in \left(3,6). The nonlinear term ∣u∣u| u| u is the so-called Coulomb critical nonlinearity because it presents a certain scaling invariance and the mountain-pass geometry cannot be established when f=0f=0. Few results are known to such case. By developing some delicate analyses and using detailed estimates, we obtain the existence of ground-states and least energy solution for the aforementioned system under some natural assumptions on ff.

Keywords

• schrödinger-poisson system
• coulomb critical exponent
• ground-state solutions
• pohozaev identity
• zero mass
• 35j20
• 35j60