Existence and nonexistence of ground state solutions for a Schrödinger–Poisson–Slater equation

Abstract

In this article, we are concerned with the Schrödinger–Poisson–Slater equation $-\mathrm{\Delta }u+(|x{|}^{\alpha -n}\ast |u{|}^p)|u{|}^{p-2}u=\mu |u{|}^{q-2}u,\mathrm{in}{\mathbb{R}}^n,$ where $n\ge 2$, $\alpha \in (0,n)$, p>1, q>1, $\mu >0$. By using the Nehari–Pohozaev manifold, we obtain the existence and nonexistence of ground state solutions results, depending on the parameters p and q. Our results extends some recent work in the literature.

Keywords:

• Schrödinger–Poisson–Slater equation
• ground state
• Nehari–Pohozaev manifold

AMS Subject Classifications:

• 35J20
• 35J61
• 35Q55

Disclosure statement

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

Additional information

Funding

Supported by National Natural Science Foundation of China [No. 11661021,11861021].