Normalized Solutions to the Kirchhoff-Choquard Equations with Combined Growth

TL;DR

This paper investigates normalized solutions to a nonlocal Kirchhoff-Choquard equation with combined growth, establishing existence, multiplicity, and ground state solutions using variational methods, including new results for supercritical cases.

Contribution

It introduces novel existence results for Kirchhoff-Choquard equations with combined growth, especially for the supercritical case where q is L^2-supercritical and p reaches the critical exponent.

Findings

Proved existence of solutions under various parameter ranges.

Established multiplicity and ground state solutions.

Analyzed asymptotic behavior of solutions.

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.