Normalized solutions for nonautonomous Schrödinger–Poisson equations

In this paper, we study the existence of normalized solutions for the nonautonomous Schrödinger–Poisson equations \(\begin{aligned} -\Delta u+\lambda u +\left( \vert x \vert ^{-1} * \vert u \vert ^{2} \right) u=A(x)|u|^{p-2}u,\quad \text {in}~\mathbb {R}^3, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+\lambda u+\left({|x|}^{-1},,\ast ,{|u|}^2\right)u=A(x){|u|}^{p-2}u,\text{in}{\mathbb{R}}^3,\end{array}$$ where \(\lambda \in \mathbb {R}\) $\lambda \in \mathbb{R}$ , \(A \in L^\infty (\mathbb {R}^3)\) $A\in L^{\infty }({\mathbb{R}}^3)$ satisfies some mild conditions. Due to the nonconstant potential A, we use Pohozaev manifold to recover the compactness for a minimizing sequence. For \(p\in (2,3)\) $p\in (2,3)$ , \(p\in (3,\frac{10}{3})\) $p\in (3,\frac{10}{3})$ and \(p\in (\frac{10}{3}, 6)\) $p\in (\frac{10}{3},6)$ , we adopt different analytical techniques to overcome the difficulties due to the presence of three terms in the corresponding energy functional which scale differently.