Normalized solutions for nonautonomous Schrödinger–Poisson equations

Abstract

In this paper, we study the existence of normalized solutions for the nonautonomous Schrödinger–Poisson equations

$$\begin{aligned} -\Delta u+\lambda u +\left( \vert x \vert ^{-1} * \vert u \vert ^{2} \right) u=A(x)|u|^{p-2}u,\quad \text {in}~\mathbb {R}^3, \end{aligned}$$

where \(\lambda \in \mathbb {R}\), \(A \in L^\infty (\mathbb {R}^3)\) satisfies some mild conditions. Due to the nonconstant potential A, we use Pohozaev manifold to recover the compactness for a minimizing sequence. For \(p\in (2,3)\), \(p\in (3,\frac{10}{3})\) and \(p\in (\frac{10}{3}, 6)\), we adopt different analytical techniques to overcome the difficulties due to the presence of three terms in the corresponding energy functional which scale differently.