Existence and asymptotics of normalized solutions for logarithmic Schr\"{o}dinger system

Abstract

This paper is concerned with the following logarithmic Schr\"{o}dinger system: $$\left\{\begin{align} \ &\ -\Delta u_1+\omega_1u_1=\mu_1 u_1\log u_1^2+\frac{2p}{p+q}|u_2|^{q}|u_1|^{p-2}u_1,\\ \ &\ -\Delta u_2+\omega_2u_2=\mu_2 u_2\log u_2^2+\frac{2q}{p+q}|u_1|^{p}|u_2|^{q-2}u_2,\\ \ &\ \int_{\Omega}|u_i|^2\,dx=\rho_i,\ \ i=1,2,\\ \ &\ (u_1,u_2)\in H_0^1(\Omega;\mathbb R^2),\end{align}\right.$$ where $\Omega=\mathbb{R}^N$ or $\Omega\subset\mathbb R^N(N\geq3)$ is a bounded smooth domain, $\omega_i\in\mathbb R$, $\mu_i,\ \rho_i>0,\ i=1,2.$ 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 Br\'{e}zis-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 Br\'{e}zis-Nirenberg problem with and without the $L^2$-mass $\int_{\Omega}|u_i|^2\,dx=\rho_i(i=1,2)$ constraint by two different methods, respectively. Our results seems to be the first result of the normalized solution of the coupled nonlinear Schr\"{o}dinger system with logarithmic perturbation.