Existence and multiplicity of normalized solutions to lower critical Choquard equation with kinds of bounded potentials

Abstract

This paper studies the existence and multiplicity of normalized solutions to the lower critical Choquard equation with a $L^2$-subcritical local perturbation and kinds of bounded potentials \begin{equation*} \begin{cases} -\Delta u+V(x)u \\ \qquad =\lambda u+ \big(I_{\alpha}\ast|u|^{({N+\alpha})/{N}}\big)|u|^{({N+\alpha})/{N}-2}u +\mu |u|^{q-2}u & \text{in } \mathbb{R}^N, \\ \displaystyle \int_{\mathbb{R}^N}|u|^2dx=a^2, \end{cases} \end{equation*} where $N\geq 1$, $\mu, a> 0$, $2< q< 2+{4}/{N}$, $\alpha\in (0,N)$, $I_{\alpha}$ is the Riesz potential, $V(x)$ is a bounded potential and $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier.