🔢 mathematics

Critical concave-convex problems in Carnot groups

This paper establishes the existence of two positive solutions for a Dirichlet problem involving concave-convex and critical nonlinearities in Carnot groups by employing a variational Perron method and carefully addressing boundary regularity challenges.

Original authors: Mattia Galeotti, Eugenio Vecchi

Published 2026-04-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mattia Galeotti, Eugenio Vecchi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a landscape architect trying to design a garden inside a very strange, curved world called a Carnot Group. This isn't a flat Euclidean garden; it's a place where movement is restricted, like driving a car that can only move forward and turn, but never slide sideways.

In this garden, you want to plant flowers (mathematical solutions) that grow according to a specific set of rules. The rules are a mix of two opposing forces:

The "Concave" Force: A gentle, sub-linear push that wants the flowers to grow easily when they are small (like a seedling needing just a little water).
The "Convex" Force: A wild, explosive push that kicks in when the flowers get big, trying to make them grow infinitely fast (like a weed taking over).

The paper by Mattia Galeotti and Eugenio Vecchi is about finding out how many distinct gardens (solutions) you can create under these rules, depending on how much "fertilizer" (a parameter called $\lambda$ ) you add.

The Main Characters

The Garden ( $\Omega$ ): A bounded, enclosed space with a defined edge (the boundary).
The Fertilizer ( $\lambda$ ): A knob you can turn. If you turn it too high, the garden explodes and nothing works. If you turn it just right, magic happens.
The Critical Exponent ( $2^*_Q$ ): This is the "speed limit" of the garden. It's the maximum rate at which the wild force can grow before the math breaks down. It's like the point where a balloon pops.

The Big Discovery: The "Sweet Spot"

The authors prove a beautiful result about the number of gardens you can build:

Too Much Fertilizer ( $\lambda > \Lambda$ ): If you add too much fertilizer, the wild force wins. The garden becomes unstable, and no solution exists. It's like trying to balance a pencil on its tip; it just falls over.
Just Enough Fertilizer ( $\lambda = \Lambda$ ): There is a critical threshold. At this exact point, you can build one garden. It's a fragile, perfect balance.
The Goldilocks Zone ( $0 < \lambda < \Lambda$ ): This is the exciting part. If you use a moderate amount of fertilizer, you don't just get one garden; you get two!
- Garden A (The Small One): A stable, calm garden where the flowers are small and happy. This is the "local minimum"—a safe valley in the landscape.
- Garden B (The Big One): A wilder, more complex garden that reaches higher. This is the "mountain peak" you have to climb over to get to.

How Did They Find the Second Garden?

In normal, flat worlds (like standard Euclidean geometry), mathematicians have a famous trick (the Brezis-Nirenberg result) to find that second garden. But in these "Carnot" worlds, the edges of the garden are tricky and jagged (called "characteristic points"). The usual tricks don't work because the math gets messy at the edges.

So, the authors had to invent a new way to find the second garden:

The Variational Perron Method (The "Safety Net"):
Imagine you are looking for the lowest point in a valley (Garden A). They used a method that builds a "safety net" of solutions. They started with a tiny seed (a subsolution) and a giant tree (a supersolution) and proved that somewhere in between, a perfect garden must exist. This guaranteed the first solution.
The "Mountain Pass" (Finding Garden B):
To find the second garden, they used a concept called the Ekeland Variational Principle.
- Think of Garden A as a deep, comfortable valley.
- To get to Garden B, you have to climb out of the valley, go over a mountain pass, and then descend into a second, different valley.
- The authors proved that this "mountain pass" exists. They showed that if you try to walk from the first garden to a state of "infinity" (where the flowers grow too big), you must cross a high ridge. The top of that ridge is the second solution.

Why Does This Matter?

This isn't just about abstract math. These "Carnot groups" describe real-world physics, like how heat moves through certain materials or how signals travel in the human brain (which has a complex, layered structure).

The Heisenberg Group (a specific type of Carnot group) is famous in physics. When the authors set the fertilizer to zero, their problem becomes the CR-Yamabe problem, which is a famous question about the shape of the universe in certain dimensions.

The Takeaway

The paper solves a puzzle that had been stuck for a long time: "Can we find two different stable states in these complex, curved worlds?"

Before: We knew one solution existed.
Now: We know that if you don't push the system too hard, you can actually find two completely different, stable realities.

It's like discovering that a single recipe, if followed with the right amount of ingredients, can result in either a delicate soufflé or a rich, dense cake, but not both at the same time—and certainly not if you add too much sugar! The authors mapped out exactly how much sugar you can add before the cake collapses, and proved that in the middle range, you have a choice between two delicious outcomes.

1. Problem Statement

The authors investigate the existence and multiplicity of positive weak solutions to a specific class of nonlinear elliptic partial differential equations (PDEs) defined on Carnot groups.

Let $G$ be a Carnot group and $\Omega \subset G$ be an open, bounded, connected set with a Lipschitz boundary. The study focuses on the Dirichlet boundary value problem $(P_\lambda)$ :

$\begin{cases} -\Delta_G u = \lambda u^q + u^{2^*_Q - 1} & \text{in } \Omega, \\ u > 0 & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases}$

where:

$-\Delta_G$ is the sub-Laplacian on the Carnot group (a sum of squares of vector fields satisfying the Hörmander condition).
$\lambda > 0$ is a parameter.
$q \in (0, 1)$ represents a concave (sublinear) nonlinearity.
$2^*_Q = \frac{2Q}{Q-2}$ is the critical Sobolev exponent, where $Q$ is the homogeneous dimension of the group. The term $u^{2^*_Q - 1}$ represents the convex (critical) nonlinearity.

This problem is a generalization of the classical Ambrosetti-Brezis-Cerami problem to the sub-Riemannian setting of Carnot groups. The interplay between the concave term (which aids existence) and the critical convex term (which causes a lack of compactness) creates a challenging variational landscape.

2. Methodology

The proof strategy relies on variational methods adapted to the specific geometric and analytic constraints of Carnot groups. The authors employ the following key techniques:

A. Variational Framework

The problem is studied in the Folland-Stein space $S^1_0(\Omega)$ , the completion of $C^\infty_0(\Omega)$ with respect to the norm $\|u\| = \|\nabla_G u\|_{L^2}$ . The associated energy functional is:
$I_\lambda(u) = \frac{1}{2}\int_\Omega |\nabla_G u|^2 - \frac{\lambda}{q+1}\int_\Omega |u|^{q+1} - \frac{1}{2^*_Q}\int_\Omega |u|^{2^*_Q}.$

B. Variational Perron Method (Existence of First Solution)

To establish the existence of a first solution, the authors adapt Struwe's variational Perron method:

Threshold Definition: They define $\Lambda = \sup \{ \lambda > 0 : (P_\lambda) \text{ admits a weak solution} \}$ .
Sub- and Supersolutions:
- A subsolution is constructed using the unique solution to the purely sublinear problem ( $-\Delta_G u = \lambda u^q$ ).
- A supersolution is constructed using the solution of the problem for a larger parameter $\lambda' > \lambda$ .
Ordering: Using a comparison principle (Lemma 2.5), they show the subsolution is less than the supersolution, allowing the application of a fixed-point argument (Lemma 2.6) to find a solution between them.

C. Local Minimizer Analysis (Handling Boundary Regularity)

A critical technical hurdle is proving that the first solution $u_\lambda$ is a local minimizer of $I_\lambda$ in the $S^1_0(\Omega)$ topology.

Challenge: In Euclidean spaces, this is often proved using the Brezis-Nirenberg result, which relies on $C^{1,\alpha}$ regularity up to the boundary. In Carnot groups, boundary regularity is delicate due to characteristic points (where the boundary is tangent to the horizontal distribution), and such regularity is not generally known.
Solution: The authors bypass the need for $C^{1,\alpha}$ regularity by following a variational approach inspired by Alama. They prove directly that $u_\lambda$ is a local minimizer in the $S^1_0$ norm by analyzing the behavior of the functional near the solution without relying on pointwise boundary estimates.

D. Second Solution via Mountain Pass / Minimax

To prove the existence of a second solution for $\lambda \in (0, \Lambda)$ , the authors use a combination of:

Ekeland's Variational Principle: Applied to the functional restricted to a specific subset of the space to find a Palais-Smale sequence.
Tarantello's Argument: They analyze two cases based on the behavior of the functional near the first solution $u_\lambda$ $u_{λ}$ :
- Case 1: The infimum of $I_\lambda$ on a sphere around $u_\lambda$ equals $I_\lambda(u_\lambda)$ . They construct a minimizing sequence in the set $\{u \ge u_\lambda\}$ and prove strong convergence.
- Case 2: The infimum is strictly greater than $I_\lambda(u_\lambda)$ . They construct a "mountain pass" geometry using a path connecting $u_\lambda$ to a scaled version of the Aubin-Talenti functions (minimizers of the Sobolev inequality).
Asymptotic Estimates: Crucial to the proof are fine asymptotic expansions of the minimizers of the Sobolev inequality (Lemma 2.3), which allow the authors to control the energy levels and ensure the Palais-Smale condition holds below the critical threshold.

3. Key Contributions

Generalization to Carnot Groups: The paper successfully extends the famous Ambrosetti-Brezis-Cerami multiplicity result from Euclidean spaces to the general setting of Carnot groups, covering the critical case $p=2$ .
Overcoming Boundary Regularity Issues: The authors provide a robust variational proof that the first solution is a local minimizer without assuming $C^{1,\alpha}$ regularity at the boundary, a significant advancement given the known difficulties with characteristic points in sub-Riemannian geometry.
Explicit Construction of Solutions: They explicitly construct the threshold $\Lambda$ and prove the existence of solutions for all $\lambda \in (0, \Lambda]$ , with multiplicity (two solutions) for $\lambda \in (0, \Lambda)$ .
Utilization of Asymptotics: The work leverages recent results on the asymptotic behavior of Sobolev minimizers in Carnot groups (for $p \neq 2$ and general $p$ ) to handle the lack of explicit formulas for minimizers in general groups (unlike the Heisenberg group where explicit formulas exist).

4. Main Results

Theorem 1.1: Let $\Omega \subset G$ be a bounded domain with Lipschitz boundary and $q \in (0, 1)$ . There exists a critical parameter $\Lambda > 0$ such that:

(A) Non-existence: For every $\lambda > \Lambda$ , problem $(P_\lambda)$ admits no weak solutions.
(B) Existence: For every $\lambda \in (0, \Lambda]$ , problem $(P_\lambda)$ admits at least one weak solution.
(C) Multiplicity: For every $\lambda \in (0, \Lambda)$ , problem $(P_\lambda)$ admits at least two distinct positive weak solutions.

5. Significance

This work is significant for the field of Sub-Riemannian Analysis and Nonlinear PDEs for several reasons:

Geometric Relevance: It addresses the CR-Yamabe problem (the case $\lambda=0$ on the Heisenberg group) as a limiting case, providing insight into how perturbations affect the existence of solutions in sub-Riemannian geometries.
Methodological Innovation: By avoiding reliance on $C^{1,\alpha}$ boundary regularity (which is an open problem at characteristic points), the authors open the door to studying critical problems in more general sub-Riemannian manifolds where such regularity is not guaranteed.
Completeness: It fills a gap in the literature regarding the concave-convex problem with critical growth in Carnot groups, complementing previous works that focused on sub-critical or purely critical cases.
Applicability: The techniques (variational Perron method combined with asymptotic analysis of Sobolev minimizers) are likely applicable to other critical problems in non-Euclidean settings, including groups of Iwasawa type and other graded Lie groups.