Normalized solutions for Kirchhoff equations with Choquard nonlinearity: mass Super-Critical Case

In the present paper, we investigated the existence of normalized solutions for the following Kirchhoff equation with Choquard nonlinearity

$ \begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\Big)\Delta u-\lambda u = \mu|u|^{q-2}u+(I_{\alpha}\ast|u|^{p})|u|^{p-2}u, \; \; \; \; x\in \mathbb{R}^{3} \end{equation*} $

with prescribed mass $ \int_{\mathbb{R}^{3}}|u|^{2}dx = c^{2} $, where $ a, b, c > 0 $, $ \mu\in\mathbb{R} $, $ \alpha\in(0, 3) $, $ \frac{10}{3}\leq q < 6 $, $ 3+\frac{\alpha}{3}\leq p < 3+\alpha $ and $ \lambda\in\mathbb{R} $ is a Lagrange multiplier. We first considered the case of $ \mu > 0 $ and obtained mountain pass type solutions. For the defocusing situation $ \mu < 0 $, we proved the existence result by constructing a minimax characterization for the energy functional. Finally, we discussed the asymptotic behavior of normalized solutions obtained above as $ b\to0^{+} $ when $ \mu > 0 $.