AI 中文总结
本文针对\\(\mathbb{R}^2\\)无界光滑区域上的质量临界Gross-Pitaevskii系统，通过计算Leray-Schauder度，证明了远离临界值的参数对应的非平凡归一化解的存在性。
AI 中文摘要
本文研究如下含质量临界指数的Gross-Pitaevskii系统的非平凡解的存在性：\\[ \left\{ \begin{array}{ll} -\Delta u_{1}+V_1(x)u_{1}=a_{1}u_{1}^3+\beta u_{1}u_{2}^2+\mu u_{1}& \hbox{ in }\Omega,\\\\ -\Delta u_{2}+V_2(x)u_{2}=a_{2}u_{2}^3+\beta u_{2}u_{1}^2+\mu u_{2}&\hbox{ in }\Omega, u_{1},u_{2}\ge 0 &\hbox{ in }\Omega, u_1=u_2=0 &\hbox{ on }\partial\Omega, \end{array}\right. \\] 满足约束\\[ \int_\Omega (u_1^2+u_2^2)=1 \\]，其中\\(\Omega\\)是\\(\mathbb{R}^2\\)中的无界光滑区域，\\(a_1、a_2、\beta\\)为正参数，\\(V_i\\)为囚禁势，\\(\mu\in\mathbb{R}\\)是未知拉格朗日乘子。我们通过计算参数\\(a_1、a_2、\beta\\)（远离某些临界值）的Leray-Schauder度来推导解的存在性。该系统可能具有半平凡解，形式为\\((u_1, 0)\\)或\\((0, u_2)\\)。本文的创新之处在于提供了机制，确保我们找到的解是非平凡的，即\\(u_1>0\\)且\\(u_2>0\\)。
英文摘要
In this paper, we study the existence of nontrivial solutions for the following Gross-Pitaevskii system involving mass-critical exponent: \[ \left\{ \begin{array}{ll} -Δu_{1}+V_1(x)u_{1}=a_{1}u_{1}^3+βu_{1}u_{2}^2+μu_{1}& \hbox{ in }Ω,\\ -Δu_{2}+V_2(x)u_{2}=a_{2}u_{2}^3+βu_{2}u_{1}^2+μu_{2}&\hbox{ in }Ω, u_{1},u_{2}\ge 0 &\hbox{ in }Ω, u_1=u_2=0 &\hbox{ on }\partialΩ, \end{array}\right. \] with the constraint \[ \int_Ω(u_1^2+u_2^2)=1, \] where $Ω$ is an unbounded smooth domain in $\mathbb{R}^2$, $a_1, a_2, β$ are positive parameters, $V_i$ are trapping potentials, and $μ\in\mathbb{R}$ is an unknown Lagrange multiplier. We derive the existence of solutions by computing the Leray-Schauder degree for the parameters $a_1,a_2, β$, which are away from some critical values. The system may have semi-trivial solutions of the form $(u_1, 0)$ or $(0, u_2)$. Our novelty is that we provide mechanisms ensuring that the solutions we find are nontrivial, i.e., $u_1>0$ and $u_2>0$.
