标题： 归一化解的存在性和渐近行为的对数薛定谔系统 显示英文标题

标题： Existence and asymptotics of normalized solutions for logarithmic Schrödinger system

摘要： 本文研究以下对数薛定谔系统： $$\left\{\begin{align} \ &\ -\Delta u_1+\omega_1u_1=\mu_1 u_1\log u_1^2+\frac{2p}{p+q}|u_2|^{q}|u_1|^{p-2}u_1,\\ \ &\ -\Delta u_2+\omega_2u_2=\mu_2 u_2\log u_2^2+\frac{2q}{p+q}|u_1|^{p}|u_2|^{q-2}u_2,\\ \ &\ \int_{\Omega}|u_i|^2\,dx=\rho_i,\ \ i=1,2,\\ \ &\ (u_1,u_2)\in H_0^1(\Omega;\mathbb R^2),\end{align}\right.$$ 其中 $\Omega=\mathbb{R}^N$ 或 $\Omega\subset\mathbb R^N(N\geq3)$ 是一个有界光滑区域， $\omega_i\in\mathbb R$， $\mu_i,\ \rho_i>0,\ i=1,2.$ 此外， $p,\ q\geq1,\ 2\leq p+q\leqslant 2^*$，其中 $2^*:=\frac{2N}{N-2}$。 通过使用Gagliardo-Nirenberg不等式和对$u\log u^2$的仔细估计，首先，我们将提供所有$2\leq p+q\leqslant 2^*$的归一化基态解存在的统一证明。其次，我们考虑归一化基态解的稳定性。最后，我们分析Sobolev亚临界情况下的解的行为，并让指数$p+q$趋近于$2^*$时取极限。值得注意的是，$(0,+\infty)$中$u\log u^2$符号的不确定性是本文的难点之一，也是我们感兴趣的原因之一。特别是，我们可以建立带有对数扰动的Brézis-Nirenberg型问题（即$p+q=2^*$）的正归一化基态解的存在性。 此外，我们的研究包括分别使用两种不同的方法，证明了带有和不带有$L^2$-质量$\int_{\Omega}|u_i|^2\,dx=\rho_i(i=1,2)$约束的对数型 Brézis-Nirenberg 问题解的存在性。 我们的结果似乎是具有对数扰动的耦合非线性 Schrödinger 系统的归一化解的第一个结果。 显示英文摘要

摘要： 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.