Normalized solutions for Kirchhoff equations with Choquard nonlinearity

This paper is concerned with the qualitative analysis of solutions for the following Kirchhoff equation with mixed nonlinearities$ \begin{align} -\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{align} $with prescribed mass $ \int_{\mathbb{R}^{3}}|u|^{2}dx = c^{2} $, where $ a, b, c>0 $, $ \mu\in\mathbb{R} $, $ \alpha\in(\frac{5}{3}, 3) $, $ 2<q<\frac{10}{3} $, $ \frac{14}{3}<p\leq3+\alpha $. We prove several existence results to the above problem, where $ \mu $ is a positive parameter. Here, $ 3+\alpha $ is the Hardy-Littlewood-Sobolev upper critical exponent, which can be seen as the Sobolev critical exponent $ 2^{\ast} $. The proof of the Palais-Smale condition is a challenge when $ p = 3+\alpha $. So we present a different method of dealing with compactness compared to the general way handling the Sobolev critical term. Finally, while for the defocusing situation $ \mu<0 $, we prove an existence result by constructing a minimax characterization for the energy functional.