Normalized ground states for a kind of Choquard–Kirchhoff equations with critical nonlinearities

Abstract

Abstract In this paper, we consider the existence of a normalized ground-state solution for the Choquard–Kirchhoff equation: { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u = λ u + μ ( I α ∗ | u | p ) | u | p − 2 u + ω | u | 4 u , in R 3 , u > 0 , ∫ R 3 | u | 2 = m 2 , in R 3 , $$ \left \{ \textstyle\begin{array}{l@{\quad}r} -(a+b\int _{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta{u}=\lambda u+\mu (I_{ \alpha}\ast |u|^{p})|u|^{p-2}u+\omega |u|^{4}u,\ \ &\text{in}\, \mathbb{R}^{3}, \\ u>0,\quad \int _{\mathbb{R}^{3}}|u|^{2}=m^{2},\,&\text{in}\, \mathbb{R}^{3}, \end{array}\displaystyle \right . $$ where a, b, m, μ, ω > 0 $\omega >0$ , p ∈ ( 2 , 7 + α 3 ) $p\in (2,\frac{7+\alpha}{3})$ , λ ∈ R $\lambda \in \mathbb{R}$ , α ∈ ( 0 , 3 ) $\alpha \in (0,3)$ , and I α $I_{\alpha}$ is a Riesz potential. Utilizing approximation methods and Schwartz symmetrization rearrangements, we establish the existence of normalized ground states for these kinds of mass-constrained Choquard–Kirchhoff problems.

Keywords

• Normalized ground-state solutions
• Choquard–Kirchhoff equation
• Approximation method
• Schwartz symmetrization rearrangements