arXiv2023
This paper is concerned with the following logarithmic Schrödinger system: $$\left\{\begin{align} \ &\ -Δu_1+ω_1u_1=μ_1 u_1\log u_1^2+\frac{2p}{p+q}|u_2|^{q}|u_1|^{p-2}u_1,\\ \ &\ -Δu_2+ω_2u_2=μ_2 u_2\log u_2^2+\frac{2q}{p+q}|u_1|^{p}|u_2|^{q-2}u_2,\\ \ &\ \int_Ω|u_i|^2\,dx=ρ_i,\ \ i=1,2,\\ \ &\ (u_1,u_2)\in H_0^1(Ω;\mathbb R^2),\end{align}\right.$$ where $Ω=\mathbb{R}^N$ or $Ω\subset\mathbb R^N(N\geq3)$ is a bounded smooth domain, $ω_i\in\mathbb R$, $μ_i,\ ρ_i>0,\ i=1,2.$ Moreover, $p,\ q\geq1,\ 2\leq p+q\leqslant 2^*$, where $2^*:=\frac{2N}{N-2}$. By using a Gagliardo-Nirenberg inequality and careful estimation of $u\log u^2$, firstly, we will provide a unified proof of the existence of the normalized ground states solution for all $2\leq p+q\leqslant 2^*$. Secondly, we consider the stability of normalized ground states solutions. Finally, we analyze the behavior of solutions for Sobolev-subcritical case and pass the limit as the exponent $p+q$ approaches to $2^*$. Notably, the uncertainty of sign of $u\log u^2$ in $(0,+\infty)$ is one of the difficulties of this paper, and also one of the motivations we are interested in. In particular, we can establish the existence of positive normalized ground states solutions for the Brézis-Nirenberg type problem with logarithmic perturbations (i.e., $p+q=2^*$). In addition, our study includes proving the existence of solutions to the logarithmic type Brézis-Nirenberg problem with and without the $L^2$-mass $\int_Ω|u_i|^2\,dx=ρ_i(i=1,2)$ constraint by two different methods, respectively. Our results seems to be the first result of the normalized solution of the coupled nonlinear Schrödinger system with logarithmic perturbation.