Nontriviality of all constrained energy minimizers

Determine whether every minimizer of the constrained energy functional for the mass-critical Gross–Pitaevskii system is nontrivial for all parameters satisfying $0<a_1,a_2<a^*$ and $0<\beta<\beta_1^*$, meaning that both components are nonzero.

Background

The paper studies normalized solutions (u1,u2)(u_1,u_2) of a two-component mass-critical Gross–Pitaevskii system on an unbounded planar domain, subject to the constraint ∫Ω(u12+u22)=1\int_\Omega (u_1^2+u_2^2)=1. A central distinction is between nontrivial solutions, for which both components are nonzero, and semi-trivial solutions of the form (u1,0)(u_1,0) or (0,u2)(0,u_2).

For parameters below the critical threshold β1∗=a∗+(a∗−a1)(a∗−a2)\beta_1^*=a^*+\sqrt{(a^*-a_1)(a^*-a_2)}, prior results establish existence of minimizers of the constrained energy. The paper notes that asymptotic analysis guarantees nontriviality when β\beta approaches β1∗\beta_1^* from below, but does not settle whether nontriviality holds throughout the entire parameter range.

References

However, it is not clear whether any minimizer of 1-30-7 is nontrivial for all $0<a_1,a_2<a*$ and $0<\beta<\beta_1*$.

1-30-7:

inf⁡(u1,u2)∈MEa1,a2,β(u1,u2).\inf_{(u_1,u_2)\in \mathcal{M} } E_{a_1,a_2,\beta}(u_1,u_2).

— Topological degree and existence of nontrivial normalized solutions for mass-critical Gross-Pitaevskii systems  (2608.19663 - Gao et al., 20 Aug 2026) in Section 1, Introduction

The existence of nontrivial solutions (even semi-trivial ones) for 1-1--1-2 is largely unknown for most parameter values $a_i$ and $\beta$, due to the lack of effective methods to handle normalized solutions with higher energy for mass-critical problems.

1-1:

{−Δu1+V1(x)u1=a1u13+βu1u22+μu1 in Ω,−Δu2+V2(x)u2=a2u23+βu2u12+μu2 in Ω,u1,u2≥0 in Ω,u1=u2=0 on ∂Ω,\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.

1-2:

∫Ω(u12+u22)=1,\int_{\Omega} (u_{1}^2+u_{2}^2)=1,

— Topological degree and existence of nontrivial normalized solutions for mass-critical Gross-Pitaevskii systems  (2608.19663 - Gao et al., 20 Aug 2026) in Section 1, Introduction