Topological degree and existence of nontrivial normalized solutions for mass-critical Gross-Pitaevskii systems

Abstract: 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 $\mathbb{R}<sup>2$, $a_1, a_2, β$ 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., $u_1&gt;0$ and $u_2&gt;0$.

• The paper develops a Leray–Schauder degree framework, supported by blow-up classification and Morse index analysis, to establish normalized solutions beyond previously studied critical parameter regimes.
• The degree equals the number of holes, k, in key interaction ranges, linking solution existence to the topology of the planar domain and yielding nontrivial states under explicit potential-gap or spectral conditions.
• The results show that a shared mass constraint can produce nontrivial solutions even when the interspecies coupling is zero, while leaving higher critical regimes and broader nontriviality conditions open.

Problem and setting

The paper studies the two-component Gross-Pitaevskii system on an unbounded smooth domain $\Omega\subset\mathbb{R}^2$ of the form $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$ (a plane with $k$ holes), with trapping potentials $V_i$ satisfying growth, monotonicity, and regularity conditions $(V_1)$–$(V_4)$:

$$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$

with $u_i\ge 0$, Dirichlet boundary conditions, the mass constraint $\int_\Omega(u_1^2+u_2^2)=1$, and $\mu$ a Lagrange multiplier. All parameters $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$0 are positive, corresponding to self-focusing intraspecies and cooperative interspecies interactions. The critical value is $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$1, where $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$2 is the unique radial solution of $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$3, and $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$4.

Prior work established existence only near minimization regimes ($$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$5, $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$6) or via blow-up constructions for $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$7 close to critical values. The paper's stated goal is to handle parameters away from these critical values, using the topology of $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$8 and properties of $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$9. The central difficulty is excluding semi-trivial solutions $k$0 or $k$1: for example, when $k$2 on $k$3, no nontrivial solution exists at all if $k$4.

Method: fixed-point reformulation and Leray-Schauder degree

Solutions are recast as fixed points of a compact operator $k$5 built from the unique solvability of an auxiliary linear problem with mass constraint; uniqueness holds provided $k$6 with strict inequality handled by an $k$7-perturbation adding $k$8 terms to each equation. The perturbed operator $k$9 is well-defined on $V_i$0, compact, and its fixed points solve a perturbed system whose solutions are automatically nontrivial.

The degree

$V_i$1

is well-defined only where solutions are uniformly bounded in $V_i$2 as $V_i$3. The paper classifies all blow-up profiles via scaling arguments: after rescaling, blow-up sequences converge locally to positive solutions of the limiting NLS system on $V_i$4, which are completely classified (up to translation) as multiples of $V_i$5 whenever $V_i$6 or $V_i$7. Matching the total mass then forces $V_i$8 in the nontrivial case, or $V_i$9 / $(V_1)$0 in the semi-trivial cases. This yields Theorem 1.1: uniform $(V_1)$1, $(V_1)$2, and $(V_1)$3 bounds for all solutions outside neighborhoods of the critical values, uniformly along homotopies $(V_1)$4.

Degree computation

For $(V_1)$5 and $(V_1)$6, nontrivial solutions reduce to symmetric pairs $(V_1)$7 solving a single scalar problem at $(V_1)$8. Blow-up solutions concentrate at non-degenerate critical points of $(V_1)$9 with $(V_4)$0, existing on one side of $(V_4)$1 determined by the sign of $(V_4)$2. A key spectral result shows that the linearized eigenvalue problem for the system has exactly $(V_4)$3 eigenvalues below one, where $(V_4)$4 is the number of negative eigenvalues of the Hessian matrix $(V_4)$5; crucially, antisymmetric modes ($(V_4)$6) have eigenvalues bounded above one, so the Morse index coincides with that of the scalar problem. A local degree formula $(V_4)$7 follows, justified by an implicit-function-theorem argument showing $(V_4)$8 for the nonlinear remainder.

Combining the degree jump across $(V_4)$9 with the Poincaré-Hopf identity $$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$0 gives the main degree formula:

Parameter regime	Degree
$$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$1, $$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$2	$$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$3
$$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$4, $$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$5	$$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$6 (number of holes)
$$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$7, $$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$8, $$-\Delta u_i + V_i(x)u_i = a_i u_i^3 + \beta u_1 u_2^2 + \mu u_i,\quad i=1,2,$$9	$u_i\ge 0$0

The appearance of $u_i\ge 0$1 — the topological invariant of $u_i\ge 0$2 — is the structural result: nonzero degree implies existence of solutions for the perturbed problems, uniformly bounded by Theorem 1.1.

Ensuring nontriviality

Degree nonvanishing alone does not prevent the limit from being semi-trivial. Two mechanisms address this:

• For $u_i\ge 0$3: if $u_i\ge 0$4, where $u_i\ge 0$5 is defined through infima of $u_i\ge 0$6 over normalized solutions of the scalar problem, a contradiction argument based on variational characterization of $u_i\ge 0$7 rules out $u_i\ge 0$8. Notably, the condition $u_i\ge 0$9 is needed precisely to make $\int_\Omega(u_1^2+u_2^2)=1$0, although the degree computation itself only requires $\int_\Omega(u_1^2+u_2^2)=1$1.
• For $\int_\Omega(u_1^2+u_2^2)=1$2: constants $\int_\Omega(u_1^2+u_2^2)=1$3 and $\int_\Omega(u_1^2+u_2^2)=1$4 built from $\int_\Omega(u_1^2+u_2^2)=1$5 yield the gap condition $\int_\Omega(u_1^2+u_2^2)=1$6, under which every nontrivial solution satisfies $\int_\Omega(u_1^2+u_2^2)=1$7. The degree restricted to the nontrivial region satisfies $\int_\Omega(u_1^2+u_2^2)=1$8; since $\int_\Omega(u_1^2+u_2^2)=1$9 and the reduction theorem gives $\mu$0, one obtains $\mu$1.

A notable consequence of the second mechanism: even at $\mu$2, where the equations decouple, the coupled mass constraint still forces a nontrivial solution — a phenomenon that fails under separate constraints $\mu$3 when $\mu$4 and $\mu$5. The authors emphasize that $\mu$6, $\mu$7, $\mu$8 are explicitly defined, so the results are not perturbative in $\mu$9.

Further results

Under the stronger assumption $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$00, the operator $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$01 is well-defined without perturbation, and two additional existence theorems hold: for any $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$02 with $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$03 defined via suprema of $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$04 norms and spectral gaps of scalar solutions; and for small $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$05 with $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$06, using thresholds involving first eigenvalues of $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$07.

Limitations and open questions

Several restrictions are explicit. The blow-up classification covers only $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$08 up to $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$09; computing the degree in $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$10 requires handling double-peaked solutions concentrating at the same point, which the authors note is substantially more complex and leave unaddressed. The classification of limiting NLS systems requires $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$11 unless $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$12, restricting the parameter ranges treated. Whether minimizers of the energy on the constraint manifold are nontrivial for all $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$13, $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$14 remains open. The nontriviality mechanisms depend on quantitative closeness conditions between $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$15 and $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$16 (or equal first eigenvalues); whether these can be weakened is not settled. Finally, the analysis is specific to dimension two and the mass-critical exponent; extensions to other dimensions would require different critical values and are not attempted here.

Conclusion

The paper develops a Leray-Schauder degree framework for mass-critical Gross-Pitaevskii systems on planar domains with holes, combining complete blow-up classification, Morse index computations for bubbling solutions, and explicit potential-gap conditions guaranteeing nontriviality. Its principal contributions are the degree formulas yielding $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$17 (the topological invariant of $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$18) away from critical parameter values, and the demonstration that nontrivial solutions persist even at $$\Omega=\mathbb{R}^2\setminus\bigcup_{i=1}^k \bar O_i$$19 under a shared mass constraint — both obtained for parameter regimes previously inaccessible to variational or perturbative methods.