Normalized Solutions to the Kirchhoff-Choquard Equations with Combined Growth

This paper is devoted to the study of the following nonlocal equation: \begin{equation*} -\left(a+b|\nabla u|_{2}^{2(θ-1)}\right) Δu =λu+α(I_μ\ast|u|^{q})|u|^{q-2}u+(I_μ\ast|u|^{p})|u|^{p-2}u \ \hbox{in} \ $\mathbb{R}$^{N}, \end{equation*} with the prescribed norm ${\int }_{{\mathbb{R}}^N}\mathrm{\mid }u{\mathrm{\mid }}^2=c^2,$ where $N\ge 3$, $0<\mu <N$, $a,b,c>0$, $1<\theta <\frac{2N-\mu }{N-2}$, $\frac{2N-\mu }{N}<q<p\le \frac{2N-\mu }{N-2}$, $\alpha >0$ is a suitably small real parameter, $\lambda \in \mathbb{R}$ is the unknown parameter which appears as the Lagrange’s multiplier and $I_{\mu }$ is the Riesz potential. We establish existence and multiplicity results and further demonstrate the existence of ground state solutions under the suitable range of $\alpha$. We demonstrate the existence of solution in the case of $q$ is $L^2-$supercritical and $p=\frac{2N-\mu }{N-2}$, which is not investigated in the literature till now. In addition, we present certain asymptotic properties of the solutions. To establish the existence results, we rely on variational methods, with a particular focus on the mountain pass theorem, the min-max principle, and Ekeland’s variational principle.