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Existence and multiplicity of solutions for a critical Grushin problem with a singular nonlinearity

This paper investigates the existence and multiplicity of positive solutions to a critical Grushin problem featuring a singular nonlinearity, analyzing how the solution behavior depends on the exponent $p$ relative to the critical Sobolev exponent associated with the Grushin operator.

Original authors: Shammi Malhotra

Published 2026-05-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shammi Malhotra

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 trying to find the perfect shape for a soap bubble, but this isn't just any bubble. It's a bubble trapped in a strange, warped room where the rules of physics change depending on where you are standing. This is the world of the Grushin operator, the mathematical tool at the heart of this paper.

Here is a simple breakdown of what the authors, led by Shammi Malhotra, are trying to solve, using everyday analogies.

1. The Setting: A Room with a "Sticky" Floor

Most math problems about shapes happen in a perfectly smooth, flat room (like a standard Euclidean space). But the Grushin operator describes a room with a "sticky" floor.

The Analogy: Imagine walking across a floor. In most places, you can move freely in any direction. But there is a specific line (or wall) in the middle of the room where the floor becomes incredibly sticky and resistant to movement in one direction.
The Math: This "stickiness" is caused by a weight factor, $|x|^{2\gamma}$ . Near this line, the math gets "degenerate," meaning standard rules don't work as well. The authors have to figure out how to solve equations in this tricky environment.

2. The Problem: A Tug-of-War with Three Forces

The paper studies a specific equation that describes a physical state (let's call it $u$ ) inside a bounded box ( $\Omega$ ). This state is being pulled by three different forces:

The Elastic Force (The Grushin Operator): This tries to smooth things out, like a stretched rubber sheet trying to return to flat.
The "Singular" Force ( $u^{-\delta}$ ): This is the tricky part. Imagine a force that gets infinitely strong the closer you get to zero. If the value of your shape gets too small, this force screams "STOP!" and pushes it back up. It's like trying to balance a pencil on its tip; the closer you get to falling over, the harder it is to stay there.
The "Critical" Force ( $\lambda u^p$ ): This is a force that grows very fast as the shape gets bigger. The authors are interested in the "Goldilocks" zone where this growth is just right (critical), too slow (subcritical), or too fast (supercritical).

The goal is to find a shape ( $u$ ) that balances all these forces perfectly while staying zero at the walls of the box.

3. The Challenge: When Math Breaks

Usually, mathematicians use a "variational" approach. Think of this as looking for the lowest point in a mountain range (the energy minimum).

The Problem: Because of the "Singular Force" (the one that screams when things get small), the "mountain" has a cliff edge. If you try to walk down to the bottom, you might fall off a cliff where the math stops making sense.
The Solution: The authors use a technique called Nonsmooth Analysis. Instead of trying to walk smoothly down the mountain, they treat the cliff edge as a valid place to stand. They use a "subdifferential" (a fancy word for a "best guess" at the slope when the ground is jagged) to find the solution.

4. The Main Discovery: One or Two Solutions?

The paper answers a big question: How many different shapes can exist that balance these forces?

The answer depends on a control knob called $\lambda$ (lambda).

The "Sweet Spot" (Small $\lambda$ ): If the external force is weak, the authors prove there are at least two different stable shapes.
- Analogy: Imagine a ball in a valley with a small bump in the middle. The ball can settle in the deep left valley or the deep right valley. Both are stable.
- One shape is "smaller" (closer to the singular force's limit), and the other is "larger."
The "Limit" (Medium $\lambda$ ): As you turn up the knob, there comes a point where the two shapes merge into one. This is the maximum limit where a solution can exist.
The "Too Much" (Large $\lambda$ ): If you turn the knob too high, the forces become unbalanced. No shape can exist that satisfies the equation. The system collapses.

5. How They Did It (The Toolkit)

To prove these results, the authors built a ladder of tools:

Subsolutions and Supersolutions: They built a "floor" (a shape that is too small) and a "ceiling" (a shape that is too big). They proved that the real answer must be sandwiched somewhere between them.
The "Blow-Up" Technique: In the critical case, they looked at what happens when the solution gets very, very sharp (like a spike). They used special "test shapes" (called bubbles) to see if the math holds up or breaks.
Strong Maximum Principle: They proved that the solution cannot be zero anywhere inside the box; it must be positive everywhere. It's like proving that if you have a soap bubble, it can't have a flat spot in the middle; it must be curved everywhere.

Summary

In plain English, this paper says:

"Even in a mathematically 'sticky' and warped room, if you have a system fighting between a singular force (that hates zero) and a growing force, you can find stable solutions. If the growing force is small enough, you will find two different stable states. If it gets too big, no solution exists. We proved this using new mathematical tools designed to handle the 'jagged' edges where standard math fails."

The paper is a theoretical triumph, establishing the existence and counting the number of solutions for a complex equation that models difficult physical environments.

Technical Summary: Existence and Multiplicity of Solutions for a Critical Grushin Problem with a Singular Nonlinearity

Problem Statement
This paper investigates the existence and multiplicity of positive weak solutions for a boundary value problem involving the Grushin operator $\Delta_\gamma$ on a smooth bounded domain $\Omega \subset \mathbb{R}^N$ ( $N=n+m$ ). The operator is defined as $\Delta_\gamma := \Delta_x + (1+\gamma)^2|x|^{2\gamma}\Delta_y$ with $\gamma > 0$ , exhibiting degeneracy along the set $\Sigma = \{0\} \times \mathbb{R}^m$ . The primary equation studied is:
$\begin{cases} -\Delta_\gamma u = \lambda u^p + u^{-\delta} & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases}$
where $\lambda > 0$ , $\delta > 0$ , and $p > 1$ . The study covers three regimes for the exponent $p$ relative to the critical Sobolev exponent $2^*_\gamma = \frac{2Q}{Q-2}$ (where $Q = m + (1+\gamma)n$ is the homogeneous dimension): subcritical ( $p < 2^*_\gamma - 1$ ), critical ( $p = 2^*_\gamma - 1$ ), and supercritical ( $p > 2^*_\gamma - 1$ ). The presence of the singular term $u^{-\delta}$ introduces significant challenges regarding regularity and integrability, particularly when $\delta \geq 3$ , where classical variational methods fail.

Methodology
The authors employ a framework based on nonsmooth analysis and variational methods, extending techniques developed by Hirano et al. [23] for the Laplacian to the Grushin setting. The core methodological steps include:

Nonsmooth Functional Framework: Due to the singularity $u^{-\delta}$ , the associated energy functional is not $C^1$ . The authors define the functional using the Fréchet subdifferential $\partial^- I$ and utilize the Cerami condition (a compactness condition weaker than the Palais-Smale condition) to handle the lack of smoothness and boundedness of minimizing sequences.
Translation of the Problem: To manage the singularity, the problem is translated by introducing $u_0$ , the unique positive solution to the purely singular problem $-\Delta_\gamma u_0 = u_0^{-\delta}$ . The original problem is reformulated in terms of $u = v - u_0$ , leading to a new equation with a modified nonlinearity $g(z, s)$ that allows for the application of variational techniques.
Sub-Supersolution Method: The existence of solutions is established using a generalized Perron-type method. The authors construct appropriate subsolutions and supersolutions within the order interval defined by the singular solution $u_0$ and auxiliary functions.
Strong Maximum Principle and Regularity: A crucial component is the derivation of a Strong Maximum Principle for the Grushin operator (Lemma 2.3) via a logarithmic estimate. This ensures the positivity of solutions. Uniform $L^\infty$ estimates and regularity results (Lemma 2.4, Lemma 5.11) are derived to control solution behavior near the degeneracy set $\Sigma$ .
Multiplicity via Linking: To prove the existence of a second solution, the authors apply a linking theorem (Theorem 2.1) adapted for nonsmooth functionals. This involves constructing a specific geometric structure in the function space where the functional exhibits a "mountain pass" geometry relative to a local minimizer.
Blow-up Analysis: In the critical case, the proof of the existence of a second solution relies on constructing a specific test function sequence (based on the extremal functions of the Sobolev inequality) to show that the energy level of the second solution lies strictly below the threshold where compactness is lost.

Key Contributions and Results
The paper establishes the following main results (Theorem 1.1):

Existence of a First Solution: For all $\delta > 0$ and $p > 1$ , there exists a threshold $\Lambda > 0$ such that for every $\lambda \in (0, \Lambda]$ , at least one positive weak solution exists.
- In the subcritical and critical cases ( $1 < p \leq 2^*_\gamma$ ), solutions belong to $C^{0,\alpha}_{loc}(\Omega) \cap L^\infty(\Omega)$ .
- In the supercritical case ( $p > 2^*_\gamma$ ), solutions belong to $L^{p+1}(\Omega)$ (and $L^\infty$ for $\lambda < \Lambda$ ).
- The first solution $u_\lambda$ is a local minimizer of the associated energy functional and is strictly increasing with respect to $\lambda$ .
Multiplicity of Solutions: For every $\lambda \in (0, \Lambda)$ , there exists a second positive weak solution distinct from the first.
- In the subcritical and critical cases, this second solution is also in $C^{0,\alpha}_{loc}(\Omega) \cap L^\infty(\Omega)$ .
- The second solution is strictly larger than the first solution ( $w > u_\lambda$ ) in $\Omega$ .
Non-existence: For $\lambda > \Lambda$ , no positive weak solutions exist (in the appropriate function spaces).
Regularity and Uniqueness of the Purely Singular Problem: The paper proves the existence and uniqueness of the solution $u_0$ to the purely singular problem $-\Delta_\gamma u_0 = u_0^{-\delta}$ , providing explicit bounds involving the solution to $-\Delta_\gamma u_1 = 1$ .

Significance and Claims
The paper claims to extend the classical Brezis-Nirenberg theory, originally formulated for the Laplacian with singular perturbations, to the degenerate Grushin operator. Its primary significance lies in:

Handling the Full Range of $\delta$ : Unlike previous works restricted to $\delta < 1$ , this analysis covers the entire range $\delta > 0$ by utilizing nonsmooth analysis tools, thereby overcoming the integrability issues associated with strong singularities.
Degeneracy Management: The work addresses the specific difficulties posed by the Grushin operator's degeneracy along $\Sigma$ , establishing a Strong Maximum Principle and regularity results tailored to this geometry.
Unified Approach: The methodology provides a unified framework to treat subcritical, critical, and supercritical exponents within the same singular setting, demonstrating that the multiplicity phenomenon (two solutions) persists even in the presence of strong singularities and operator degeneracy.

The authors position their work as a rigorous extension of existing results on the Laplacian (citing [23], [2], [4]) to the Grushin setting, filling a gap in the literature regarding critical singular problems for degenerate elliptic operators.