Nonexistence of ground state sign-changing solutions for autonomous Schrödinger–Poisson system with critical growth

Abstract

This paper is concerned with the following Schrödinger–Poisson system $\{\begin{array}{ll}-\mathrm{\Delta }u+u+\varphi u=|u{|}^{p-1}u+|u{|}^{4}u,& x\in {\mathbb{R}}^3,\\ -\mathrm{\Delta }\varphi =u^2,& x\in {\mathbb{R}}^3,\end{array}$where 3<p<5. With the help of an odd Nehari manifold and ‘energy doubling’ property, we prove the nonexistence of ground state sign-changing solutions on $H^1({\mathbb{R}}^3)$. In this sense, our result explains why the existing literature can only consider the existence of the ground state sign-changing solutions in the radial Sobolev space $H_r^1({\mathbb{R}}^3)$.

Keywords:

• Schrödinger–Poisson system
• critical growth
• sign-changing solution

2010:

• 35A15
• 35J20
• 35J50

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This Project supported by National Natural Science Foundation of China [grant number 12171039]