Normalized bound states for the Choquard equations in exterior domains

In this paper, we investigate the following nonlinear Choquard equation with prescribed \(L^2\) $L^2$ -norm constraint \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda u+(|x|^{-1} *|u|^2)u &{}\text{ in }\ {\Omega }, \\ u=0&{}\text{ on }\ {\partial \Omega }, \\ \int \limits _\Omega |u|^2{\textrm{d}}x=a^2,\\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u=\lambda u+(|x{|}^{-1}\ast |u{|}^2)u\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \\ \underset{\mathrm{\Omega }}{\int }{|u|}^2\text{d}x=a^2,\hfill \end{array}\right)\end{array}$$ where \(a>0\) $a>0$ , \(\lambda \in \mathbb R\) $\lambda \in \mathbb{R}$ appears as an unknown Lagrange multiplier and \(\Omega \subset \mathbb R^3\) $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is an exterior domain with smooth boundary \(\partial \Omega \ne \emptyset \) $\partial \mathrm{\Omega }\ne \mathrm{\varnothing }$ such that \(\mathbb R^3\backslash \Omega \) ${\mathbb{R}}^3\backslash \mathrm{\Omega }$ is bounded. By using the splitting lemma for the unconstrained problem in exterior domains, we prove the compactness of Palais–Smale sequences corresponding to the above problem at higher energy levels. Then combining the barycentric function and Brouwer degree theory, we establish the existence of positive normalized bound states for any \(a>0\) $a>0$ provided that \(\mathbb R^3\backslash \Omega \) ${\mathbb{R}}^3\backslash \mathrm{\Omega }$ is contained in a small ball and explain that the restriction on domain \(\Omega \) $\mathrm{\Omega }$ can be equivalently transferred to a. In addition, under the radial setting of domain \(\Omega \) $\mathrm{\Omega }$ , we use genus theory to obtain the existence and multiplicity of radial normalized solutions for any \(a>0\) $a>0$ . Finally, we point out that the main results can be extended to a general mass subcritical Choquard equation.