Existence and regularity for an entire Grushin-Choquard equation

This paper establishes the existence of a mountain pass solution and proves its global $L^q$ integrability and local Hölder continuity for an entire Choquard equation involving the Grushin operator on $\mathbb{R}^N$.

The Gist

Imagine you are trying to find a specific, perfect shape for a flexible sheet of rubber. This sheet represents a mathematical solution to a complex physics problem. The problem involves two main forces: one trying to smooth the sheet out (like a spring), and another trying to pull it based on how the whole sheet is shaped (a long-range interaction).

This paper, written by Federico Bernini and Paolo Malanchini, is about finding that perfect shape on a very strange, warped piece of rubber that doesn't behave like normal space.

Here is the breakdown of their journey, explained simply:

1. The Strange Landscape: The "Grushin" World

Usually, when mathematicians study these shapes, they imagine a flat, uniform world (like a standard sheet of graph paper). In this world, moving left, right, up, or down feels exactly the same.

But in this paper, the authors are working in a Grushin world. Imagine a landscape where the ground is sticky and heavy in some places and slippery in others.

• The Analogy: Think of walking through a field. In the "x" direction, the grass is short and easy to walk through. But in the "y" direction, the grass gets taller and thicker the further you get from the center line. The further you go out, the harder it is to move sideways.
• The Problem: Because the ground is so uneven (mathematicians call this "degenerate"), the usual rules for finding the perfect shape break down. You can't just use the standard tools because the "map" of the world changes depending on where you are.

2. The Goal: The "Choquard" Equation

The specific equation they are solving is called a Choquard equation.

• The Analogy: Imagine a group of people holding hands in a giant circle. Each person is being pulled toward the center by a spring (the first part of the equation). But, they are also being pulled by a "gravity" created by everyone else in the circle. If the group gets too big, the pull changes.
• The authors want to prove that there is a stable, perfect circle (a solution) that can exist in this strange, sticky Grushin world, even though the world stretches infinitely in all directions.

3. The Challenge: The "Lost" Solution

In normal math problems, if you look for a solution on an infinite sheet, you often run into a problem called loss of compactness.

• The Analogy: Imagine you are looking for a specific bird in a forest. If the forest is small, you can find it. But if the forest is infinite, the bird could be hiding anywhere, or it could be running away forever. You can't pin it down.
• In the Grushin world, this is even worse because the "forest" itself is warped. The bird (the solution) could slip away into the "sticky" parts of the grass where the usual math tools can't grab it.

4. The Solution: The "Symmetry" Trick

To catch the bird, the authors used a clever trick called Symmetric Criticality.

• The Analogy: Instead of looking for the bird anywhere in the infinite, messy forest, they decided to only look in a perfectly symmetrical clearing in the middle. They said, "Let's only look for shapes that look the same if you spin them or flip them."
• By restricting their search to these perfectly symmetrical shapes, the "infinite forest" suddenly becomes manageable. The symmetry acts like a fence that keeps the solution from running away.
• Once they found a solution in this symmetrical clearing, they used a mathematical principle (the Principle of Symmetric Criticality) to prove that this solution actually works for the entire messy, infinite forest, not just the clearing.

5. The Result: A Smooth, Perfect Shape

After proving the solution exists, they had to check if it was a "good" solution.

• The Question: Is the shape jagged and broken, or is it smooth?
• The Finding: They proved that the solution is smooth (mathematically, it belongs to a class of functions that are continuous and don't have sharp spikes).
• The Analogy: They showed that even though the ground is sticky and weird, the rubber sheet settles into a perfectly smooth, gentle curve. It doesn't tear or have sharp edges.

Summary

In short, Bernini and Malanchini solved a puzzle about finding a stable shape in a warped, infinite universe where the usual rules of geometry don't apply.

1. They acknowledged the world was too weird for standard tools.
2. They used symmetry as a safety net to catch the solution.
3. They proved the solution exists and is smooth and well-behaved.

This is important because it helps us understand how complex physical systems (like quantum particles or fluid flows) behave in environments that aren't uniform, which is a common reality in the real world.

Technical Summary

Here is a detailed technical summary of the paper "Existence and Regularity for an Entire Grushin-Choquard Equation" by Federico Bernini and Paolo Malanchini.

1. Problem Statement

The authors investigate the existence and regularity of solutions to the following Grushin-Choquard equation defined on the entire Euclidean space $\mathbb{R}^N$ ($N \ge 3$):

$$-\Delta_\gamma u + u = \left( d(z)^{-\mu} * |u|^p \right) |u|^{p-2}u, \quad \text{in } \mathbb{R}^N$$

Key Components:

• Operator: $\Delta_\gamma$ is the Grushin operator, defined as $\Delta_\gamma u = \Delta_x u + |x|^{2\gamma}\Delta_y u$, where $z=(x,y) \in \mathbb{R}^m \times \mathbb{R}^\ell$ ($m+\ell=N$) and $\gamma \ge 0$. This operator is degenerate on the subspace $\Sigma = \{0\} \times \mathbb{R}^\ell$.
• Distance: $d(z) = (|x|^{2(\gamma+1)} + |y|^2)^{\frac{1}{2(\gamma+1)}}$ is the Grushin distance, associated with the anisotropic dilations $\delta_t(x,y) = (tx, t^{\gamma+1}y)$.
• Nonlinearity: The right-hand side is a non-local term involving the convolution of $|u|^p$ with the Riesz potential $d(z)^{-\mu}$ (where $0 < \mu < N_\gamma$), multiplied by$|u|^{p-2}u$.
• Parameters: $N_\gamma = m + (1+\gamma)\ell$ is the homogeneous dimension. The exponent $p$ must satisfy specific bounds related to the Hardy-Littlewood-Sobolev (HLS) inequality adapted to the Grushin setting.

2. Methodology

The authors employ a variational approach combined with symmetry arguments and regularity bootstrapping.

A. Functional Setting

• Space: The problem is set in the Grushin-Sobolev space $H^1_\gamma(\mathbb{R}^N)$, equipped with the norm $\|u\|_\gamma = (\int_{\mathbb{R}^N} (|\nabla_\gamma u|^2 + |u|^2) dz)^{1/2}$.
• Energy Functional: The associated energy functional is:
  $$E_\gamma(u) = \frac{1}{2}\int_{\mathbb{R}^N} (|\nabla_\gamma u|^2 + |u|^2) dz - \frac{1}{2p} \int_{\mathbb{R}^N} (d(z)^{-\mu} * |u|^p)|u|^p dz$$
• Compactness Issue: A major challenge in entire space problems is the lack of compactness due to the non-compactness of the embedding $H^1_\gamma(\mathbb{R}^N) \hookrightarrow L^q(\mathbb{R}^N)$. Unlike the standard Laplacian, the Grushin operator lacks translation invariance in the $x$-directions due to the weight $|x|^{2\gamma}$, making standard Concentration-Compactness arguments difficult to apply directly.

B. Strategy for Existence

To recover compactness, the authors restrict the search to the subspace of radially symmetric functions (symmetric in both $x$ and $y$ variables):
$$H^1_{\gamma, \text{rad}_{x,y}}(\mathbb{R}^N) = \{ u \in H^1_\gamma(\mathbb{R}^N) : u(x,y) = u(|x|, |y|) \}$$

• Compact Embedding: They utilize a known result (Lemma 2.1) stating that the embedding of this radial subspace into $L^q(\mathbb{R}^N)$ is compact for $q \in (2, 2^*_\gamma)$.
• Mountain Pass Theorem: They verify the geometric conditions of the Mountain Pass Theorem (Ambrosetti-Rabinowitz) on this subspace.
• Symmetric Criticality: Once a critical point $u$ is found in the radial subspace, they apply the Principle of Symmetric Criticality (Palais) to conclude that $u$ is also a critical point of the functional over the entire space $H^1_\gamma(\mathbb{R}^N)$, thus solving the original problem without symmetry constraints.

C. Strategy for Regularity

To prove that weak solutions are bounded and continuous, the authors use a two-step iterative process:

1. Brezis-Kato Argument (Truncation): They use a truncation technique ($u_M = \min\{u, M\}$) combined with the HLS inequality to establish that the solution belongs to $L^q(\mathbb{R}^N)$ for all $q \in [2, \infty)$. This involves a bootstrap argument where the integrability exponent is iteratively increased.
2. $L^\infty$ Bound and Hölder Continuity:
   • They first prove the non-local term $K(z) = d(z)^{-\mu} * |u|^p$ is in $L^\infty(\mathbb{R}^N)$.
   • This reduces the equation to a form $-\Delta_\gamma u = \Psi(z, u)$ with bounded coefficients.
   • They employ De Giorgi's iteration technique to prove $u \in L^\infty(\mathbb{R}^N)$.
   • Finally, local Hölder continuity ($C^{0,\alpha}_{\text{loc}}$) is deduced from the non-homogeneous Harnack inequality for $X$-elliptic operators.

3. Key Contributions and Results

Theorem 1.2 (Existence)

Under the condition that $\mu \in (0, N_\gamma)$ and the exponent $p$ satisfies:
$$\frac{N_\gamma - 2}{2N_\gamma - \mu} < \frac{1}{p} < \frac{N_\gamma}{2N_\gamma - \mu}$$
There exists a non-trivial weak solution $u \in H^1_\gamma(\mathbb{R}^N)$ to the Grushin-Choquard equation.

• Significance: This is the first existence result for this specific equation on the entire space $\mathbb{R}^N$ using variational methods, overcoming the lack of translation invariance.

Theorem 1.3 (Regularity)

Assuming $\mu > 4$ (a technical condition for the specific iteration used, though the authors note standard Laplacian cases do not require this) and $p$ satisfying the existence condition:

• Any weak solution $u$ belongs to $L^q(\mathbb{R}^N)$ for all $q \in [2, \infty]$.
• The solution is locally Hölder continuous: $u \in C^{0,\alpha}_{\text{loc}}(\mathbb{R}^N)$ for some $\alpha \in (0, 1)$.

4. Significance and Context

• Bridging Gaps: The paper addresses a significant gap in the literature regarding nonlinear Grushin-type PDEs on unbounded domains. Previous works largely focused on linear aspects (heat kernels, unique continuation) or nonlinear problems on bounded domains where compactness is easier to recover.
• Handling Degeneracy: The work demonstrates how to adapt variational methods to operators with degenerate coefficients and anisotropic scaling, specifically dealing with the breakdown of standard translation invariance.
• Methodological Innovation: The combination of restricting to radial functions to recover compactness and then lifting the result to the full space via the Principle of Symmetric Criticality provides a robust template for studying other entire Grushin equations.
• Regularity in Subelliptic Settings: The regularity proof adapts classical tools (Brezis-Kato, De Giorgi) to the Grushin geometry, establishing that despite the degeneracy, solutions to this non-local equation possess high regularity properties similar to those in the classical Euclidean setting.

In summary, this paper establishes the existence of ground-state solutions for a non-local, degenerate elliptic equation on $\mathbb{R}^N$ and proves their global boundedness and local regularity, advancing the understanding of nonlinear analysis in sub-Riemannian geometries.