Existence of normalized solutions for a class of Kirchhoff–Schrödinger–Poisson equations in $\mathbb {R}^3$

Abstract

In this paper, we study the existence of normalized solutions for the following Kirchhoff–Schrödinger–Poisson equation,

$$\begin{aligned} -\Big (a+b\int _{\mathbb {R}^3}|\nabla u|^2\mathrm{{d}}x\Big )\Delta u+(|u|^{-1}*u^2)u=f(u)+\lambda u,~~\lambda \in \mathbb {R}, \end{aligned}$$

where $a>0$, $b>0$, $f\in C(\mathbb {R},\mathbb {R})$ satisfies more general conditions and there exists a constant $p\in (\frac{14}{3},6)$, such that $\lim _{|t| \rightarrow 0 }\big [f(t)t-pF(t)\big ]=0$, where $F(t)=\int _0^tf(s)\mathrm{{d}}s$. Some new analytical techniques are introduced to overcome the difficulties due to the presence of four terms in the corresponding energy functional which scale differently.