Existence and asymptotics of normalized solutions for the logarithmic Schrödinger system

This paper is concerned with the following logarithmic Schrödinger system

\(\left\{ {\matrix{{ - \Delta {u_1} + {\omega _1}{u_1} = {\mu _1}{u_1}\log \,u_1^2 + {{2p} \over {p + q}}|{u_2}{|^q}|{u_1}{|^{p - 2}}{u_1},} \hfill \cr { - \Delta {u_2} + {\omega _2}{u_2} = {\mu _2}{u_2}\log \,u_2^2 + {{2q} \over {p + q}}|{u_1}{|^p}|{u_2}{|^{q - 2}}{u_2},} \hfill \cr {\int_\Omega {|{u_i}{|^2}dx = {\rho _i},\,\,\,\,\,i = 1,2,} } \hfill \cr {({u_1},{u_2}) \in H_0^1(\Omega ;{\mathbb{R}^2}),} \hfill \cr } } \right.\) $$\{\begin{array}{l}-\Delta u_1+{\omega }_1{u}_1={\mu }_1{u}_1\log u_1^2+\frac{2p}{p+q}\mid u_2{\mid }^q\mid u_1{\mid }^{p-2}{u}_1,\hfill \\ -\Delta u_2+{\omega }_2{u}_2={\mu }_2{u}_2\log u_2^2+\frac{2q}{p+q}\mid u_1{\mid }^p\mid u_2{\mid }^{q-2}{u}_2,\hfill \\ {\displaystyle {\int }_{\Omega }\mid u_i{\mid }^{2}dx={\rho }_i,i=1,2,}\hfill \\ (u_1,u_2)\in H_0^1(\Omega ;{ℝ}^2),\hfill \end{array}$$

where Ω = ℝ^N or Ω ⊂ ℝ^N (N ⩾ 3) is a bounded smooth domain, and ω_i ∈ ℝ, μ_i, ρ_i > 0 for i = 1, 2. Moreover, p,q ⩾ 1, and 2 ⩽ p + q ⩽ 2*, where \({2^ * }: = {{2N} \over {N - 2}}\) $$2^{\ast }:=\frac{2N}{N-2}$$ . By using a Gagliardo-Nirenberg inequality and a careful estimation of u log u², firstly, we provide a unified proof of the existence of the normalized ground state solution for all 2 ⩽ p + q ⩽ 2*. Secondly, we consider the stability of normalized ground state solutions. Finally, we analyze the behavior of solutions for the Sobolev-subcritical case and pass to the limit as the exponent p + q approaches 2*. Notably, the uncertainty of the sign of u log u² in (0, +∞) 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 state 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 L²-mass constraint ∫_Ω∣u_i∣²dx = ρ_i (i = 1, 2) by two different methods, respectively. Our results seem to be the first result of the normalized solution of the coupled nonlinear Schrödinger system with logarithmic perturbations.