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

Note that the limit case is left open.

— The nonlinear Schrödinger equation on products of $\mathbb{R}^N$ and compact metric graphs  (2609.18698 - Soave et al., 16 Sep 2026) in Remark immediately following Proposition 6.1 (labelled “prop mu1”), Section 6, “Local minimality of the trivial solution”

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