Planar doubling nodal solutions to the Yamabe equation with maximal rank

Abstract: This article constructs two families of nodal solutions to the Yamabe equation, each concentrating along two planar circles. One family is conformally equivalent to the one previously obtained by Medina--Musso. The second family is a twisted variant of the first; it is new and is derived from ansatzes that are not Kelvin invariant, in contrast to a standard assumption in earlier works. In addition, in dimension 3, these solutions attain maximal rank. By means of a continuous family of conformal transformations, we then analyze the interaction of the two circles, which display a crossing phenomenon reminiscent, in some sense, of leap-frogging behavior in vortex dynamics.

• The paper presents a unified approach to construct planar doubling nodal solutions with maximal rank for both Kelvin-invariant and twisted configurations.
• It employs an inner-outer gluing method and detailed asymptotic analysis to control bubble interactions and error terms as the number of bubbles increases.
• The work extends maximal-rank solutions across dimensions by relaxing Kelvin invariance, offering novel insights into energy-critical elliptic equations.

Planar Doubling Nodal Solutions to the Yamabe Equation with Maximal Rank

Introduction and Motivation

This paper constructs new families of sign-changing solutions with maximal conformal rank to the Yamabe equation on $\mathbb{R}^n$. The core analytic object of study is the energy-critical semilinear elliptic equation

$$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$

where $u$ belongs to the usual completion $\mathcal D^{1,2}(\mathbb{R}^n)$.

Classical positive solutions to this equation are explicit, forming the Aubin–Talenti bubbles, and any such solution is conformally equivalent to a standard spherical solution. The understanding of sign-changing (nodal) solutions, and especially their structures under conformal symmetries, is less complete. The notion of maximal rank is central: a solution $Q$ has maximal rank if the tangent space generated by the action of the Möbius group $\mathrm{Mob}(n)$ at $Q$ achieves the dimension of the full group.

Earlier constructions of entire nodal solutions either lacked maximal rank in some dimensions, or relied crucially on Kelvin (inversion) symmetry in their gluing procedures. Notably, Medina, Musso, and Wei produced two types of maximal-rank nodal solutions: one by "doubling" the equator (a circle) in dimension $n=3$ [MEDINA2021], and another by constructing a product configuration in even dimensions [MEDINA2019]. However, these constructions are essentially Kelvin-invariant.

The main contributions of the present work are twofold:

1. Unified approach to maximal-rank solutions for both even and odd dimensions with configurations based on planar circles, applicable regardless of parity;
2. Relaxation of Kelvin invariance in the gluing construction, yielding a genuinely new twisted (non-Kelvin-invariant) family of nodal solutions.

Main Results

Families of Planar Doubling Solutions

The authors construct, for large integer $k$, two families of finite-energy, sign-changing solutions to the Yamabe equation concentrating along two coplanar circles in the $x_1x_2$-plane. The general ansatz is

$$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$0

where $$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$1 is the Aubin–Talenti bubble, $$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$2 are small scales, and $$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$3 are centers located, for each $$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$4, at the vertices of two concentric circles of radii $$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$5 and $$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$6 with precise rotations:

• $$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$7: circles are in-phase (Kelvin-invariant configuration).
• $$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$8: circles shifted by $$- \Delta u = \gamma |u|^{p-1} u \quad \text{in } \mathbb{R}^n, \qquad \gamma = \frac{n(n-2)}{4}, \quad p = \frac{n+2}{n-2}, \quad n \geq 3,$$9 (twisted, non-Kelvin-invariant configuration).

The main technical advance is the control of the asymptotic interaction geometry and the error analysis in the (non-)invariant case, enabling the construction for both $u$0.

For $u$1, these solutions attain the dimension $u$2 for the Möbius tangent space, equaling the dimension of $u$3, hence maximal conformal rank.

Gluing and Conformal Analysis

The authors establish the existence of the solutions via an inner-outer gluing method:

• They construct a matched asymptotic expansion with a central bubble, $u$4 inner satellites, and $u$5 outer satellites.
• Both inner and outer problems are solved, and all relevant error terms are decomposed into symmetry classes to exploit the group structure.
• An explicit finite-dimensional reduced problem for the gluing parameters yields solutions whose configuration is controlled as $u$6.

A conformal deformation parameterized by points on the $u$7-axis (the $u$8 family) reveals a "crossing" or "leapfrogging" phenomenon: under this Möbius flow, the two circles approach, cross, and swap their radii and heights above/below the reference plane. This geometric evolution clarifies how the new planar doubling families relate (or do not relate) to previously known solutions under conformal equivalence.

A key result is that the twisted family ($u$9) is not conformally equivalent to the Kelvin-invariant family ($\mathcal D^{1,2}(\mathbb{R}^n)$0) or to any solution constructed previously when $\mathcal D^{1,2}(\mathbb{R}^n)$1 is large. This is established by a classification of conformal images of coplanar circle configurations.

Technical Novelties

• Non-Kelvin-invariant gluing: Prior gluing schemes for such problems enforced Kelvin symmetry. The new twisted construction demonstrates that solutions with maximal rank can exist without this invariance, by introducing a nontrivial rotation between the inner and outer sets of bubbles.
• Maximal-rank solution in all dimensions: By appropriately distributing bubbles in pairs of planes ($\mathcal D^{1,2}(\mathbb{R}^n)$2, $\mathcal D^{1,2}(\mathbb{R}^n)$3), the method generalizes to higher even and odd $\mathcal D^{1,2}(\mathbb{R}^n)$4, yielding maximal-rank families for all $\mathcal D^{1,2}(\mathbb{R}^n)$5.
• Precise interaction and asymptotic formulas: The detailed summation and expansion formulas for bubble interactions yield explicit scaling laws for the small parameters as $\mathcal D^{1,2}(\mathbb{R}^n)$6.

Implications and Future Directions

The existence of planar doubling nodal solutions, including non-Kelvin-invariant and generically non-degenerate families, advances the structural theory of critical elliptic equations and their solution moduli in high dimensions. Practically, these configurations offer new energy profiles and symmetry-breaking templates for variational and dynamical problems, notably in geometric analysis and mathematical physics (for example, the energy-critical focusing wave equation).

The methodology demonstrates the flexibility of gluing constructions beyond standard symmetry constraints, suggesting that further non-conformally-equivalent solution families exist even for other nonlinear conformally invariant equations (e.g., higher order or system analogues).

The question of non-degeneracy of these new solutions (i.e., kernel equality, no extra Jacobi fields) remains open, as in much of the existing literature. Resolving this would clarify stability and uniqueness aspects and their role in blow-up analysis for geometric and wave equations.

Conclusion

This work constructs two new families of entire sign-changing nodal solutions to the Yamabe equation on $\mathcal D^{1,2}(\mathbb{R}^n)$7:

• Both concentrate along pairs of planar circles and, for $\mathcal D^{1,2}(\mathbb{R}^n)$8, possess maximal Möbius rank.
• One is Kelvin-invariant (conformally equivalent to existing solutions), while the other is a genuinely new, twisted, planar configuration not obtainable via conformal symmetries.
• The approach unifies maximal-rank constructions across dimensions and fundamentally extends known methods by weakening symmetry assumptions.

The results reinforce the richness of the moduli of nodal solutions for critical equations and provide a foundation for further exploration into geometric and analytic properties of such high-rank solutions.

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Reference: “Planar doubling nodal solutions to the Yamabe equation with maximal rank” (2604.02978)