Normalized Ground State Solutions to HLS Upper Critical Choquard Equation With Mass Critical or Supercritical and Sobolev Critical Growth

Abstract

This paper is concerned with the following HLS upper critical Choquard equation with mass critical or supercritical and Sobolev critical growth $\left(\begin{array}{c}-\Delta u=\lambda u+\mu |u{|}^{p-2}u+|u{|}^{4}u+(I_2\ast |u{|}^5\left)\right|u{|}^{3}u\text{in}{ℝ}^3,\\ {\int }_{{ℝ}^3}{u}^{2}dx=c\end{array}\right),$ where $\mu ,c>0,\frac{10}{3}\le p<6,\lambda \in ℝ$ is a Lagrange multiplier and $I_2$ is the Riesz potential. The novelty of this paper is that, utilizing the classical strategy due to Jeanjean (Nonlinear Analysis: Theory, Methods & Applications, 28(1997), 1633-1659) and after some subtle energy estimate, we show the existence of mountain pass type normalized ground state solutions for any $\mu >0$. In this sense, the novelty of this paper includes three aspects: when $\frac{10}{3}<p<6$, we do not need to add any constraints to the parameter $\mu >0$; meanwhile, the mountain pass type solution happens to be one ground states; in addition, the $L^2$-critical perturbation, that is, $p=\frac{10}{3}$ is considered, which generalize and improve the recent results.