Normalized Solutions to the Kirchhoff-Choquard Equations with Combined Growth

Abstract

This paper is devoted to the study of the following nonlocal equation: \begin{equation*} -\left(a+b\|\nabla u\|_{2}^{2(\theta-1)}\right) \Delta u =\lambda u+\alpha (I_{\mu}\ast|u|^{q})|u|^{q-2}u+(I_{\mu}\ast|u|^{p})|u|^{p-2}u \ \hbox{in} \ \mathbb{R}^{N}, \end{equation*} with the prescribed norm $ \int_{\mathbb{R}^{N}} |u|^{2}= c^2,$ where $N\geq 3$, $0<\mu<N$, $a,b,c>0$, $1<\theta<\frac{2N-\mu}{N-2}$, $\frac{2N-\mu}{N}<q<p\leq \frac{2N-\mu}{N-2}$, $\alpha>0$ 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.