Multiple Normalized Solutions for Choquard equations involving Kirchhoff type perturbation

Abstract

In this paper we study the existence of critical points of the $C^1$ functional \begin{equation*} E(u)=\frac{a}{2}\int_{\bbrn}|\nabla u|^2dx+\frac{b}{4}\bigg(\int_{\bbrn}|\nabla u|^2dx\bigg)^2-\frac{1}{2p}\int_{\bbrn}(I_\alpha*|u|^p)|u|^pdx \end{equation*} under the constraint \begin{equation*} S_c=\bigg\{u\in H^1(\bbrn)\ \bigg\vert\ \int_{\bbrn}|u|^2dx=c^2\bigg\}, \end{equation*} where $a> 0$, $b> 0$, $N\geq3$, $\alpha\in(0,N)$, $ (N+\alpha)/{N}< p< (N+\alpha)/$ $(N-2)$ and $I_{\alpha}$ is the Riesz Potential. When $p$ belongs to different ranges, we obtain the threshold values separating the existence and nonexistence of critical points of $E$ on $S_c$. We also study the behaviors of the Lagrange multipliers and the energies corresponding to the constrained critical points when $c\to 0$ and $c\to +\infty$, respectively.