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Exact number of positive solutions and existence of sign-changing solutions with prescribed mass for NLS on bounded domains

This paper establishes the exact number of positive solutions and the existence of sign-changing solutions with prescribed mass for the nonlinear Schrödinger equation on bounded domains across $L^2$ -subcritical, critical, and supercritical regimes, while also characterizing the asymptotic behavior of the associated parameters and energies as the mass approaches zero or infinity.

Original authors: Linjie Song, Wenming Zou

Published 2026-03-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Linjie Song, Wenming Zou

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 chef trying to bake the perfect cake. But there's a catch: you must use exactly 500 grams of flour (this is your "prescribed mass"). You can't use more, you can't use less.

Now, imagine the cake batter is a mathematical wave (a solution to an equation) and the oven is a specific room (a "bounded domain"). The goal of this paper is to figure out: How many different cakes can we bake with exactly 500 grams of flour, and what do they look like?

The authors, Linjie Song and Wenming Zou, are investigating a complex recipe known as the Nonlinear Schrödinger Equation (NLS). In the real world, this equation describes how light travels through fiber optics or how atoms behave in a quantum cloud. But in their math world, they are asking: "If we fix the total amount of 'stuff' (mass), how many stable shapes can this stuff take?"

Here is the breakdown of their discovery, using simple analogies:

1. The Three Types of Recipes (The Power of the Nonlinearity)

The paper looks at three different "flavors" of the recipe, depending on how the ingredients interact (the exponent $p$ ):

The "Gentle" Recipe ( $L^2$ -subcritical): The ingredients interact softly.
- The Discovery: If you have a tiny amount of flour (small mass), you can bake infinitely many different cakes. Some are simple, some are wild and complex. As you make the flour amount smaller and smaller, the cakes get flatter and flatter, and the "heat" (energy) needed to bake them drops to zero.
The "Critical" Recipe ( $L^2$ -critical): The ingredients interact at a perfect tipping point.
- The Discovery: Similar to the gentle recipe, but only if your flour amount is small enough. You still get infinitely many cakes, but if you try to use too much flour, the recipe breaks, and no cake can be made.
The "Explosive" Recipe ( $L^2$ -supercritical): The ingredients interact violently.
- The Discovery: This is the tricky one. If you use a lot of flour, the batter explodes, and you can't bake a cake at all. However, if you use a very small amount of flour, you can bake as many cakes as you want.

2. The Two Types of Cakes: Positive vs. Sign-Changing

In math, a "positive" solution is like a cake that is entirely above the table (all positive numbers). A "sign-changing" solution is like a cake that has a hill and a valley—it goes above and below the table.

The Positive Cakes:
- In the "Explosive" recipe (supercritical) with a round room (a ball), the authors proved something very specific: There are exactly two positive cakes you can bake with a tiny amount of flour.
- Cake A: It's a small, gentle hill. As you reduce the flour, this cake gets flatter and the "heat" (energy) drops to zero.
- Cake B: It's a massive, towering mountain. As you reduce the flour, this mountain gets taller and taller, and the heat required to bake it goes to infinity.
- Why this matters: Finding these two specific cakes was the key to unlocking the next mystery.
The Sign-Changing Cakes (The Main Event):
- The authors wanted to find a cake that has both a hill and a valley (a sign-changing solution) that is massive (high energy) and requires infinite heat.
- The Challenge: Usually, when you look for these wild, complex shapes, the math gets messy, and the solutions might disappear or turn into simple positive cakes.
- The Breakthrough: By first proving exactly how the two "Positive Cakes" behave (the two hills), they built a bridge. They used a technique called "Descending Flow" (imagine a hiker walking down a mountain) to navigate the landscape of possibilities.
- The Result: They found a third cake (a sign-changing one) that is huge and wild. As the flour amount ( $\mu$ ) goes to zero, this cake becomes infinitely tall and requires infinite energy. It's a "monster" solution that exists only when the mass is tiny.

3. The Tools They Used (The Kitchen Gadgets)

To find these solutions, the authors invented and used some clever mathematical tools:

The "Genus" Theory: Imagine you are trying to find the highest point on a map. If the map is a simple hill, it's easy. But if the map is a complex mountain range with many peaks, you need a way to count the peaks. "Genus" is a topological tool that counts how many "holes" or "peaks" a shape has. They used this to prove there must be many different solutions.
The "Descending Flow": Imagine you are on a mountain and you want to find a specific valley. Instead of guessing, you just start walking downhill. The authors created a special "downhill path" that guarantees you won't accidentally walk into a "positive-only" zone (where the cake is all above the table). This forced them to find the sign-changing solution.
The "Pseudo-Gradient": In a normal kitchen, you follow a recipe. In this math world, the "gradient" (the direction of steepest descent) sometimes points the wrong way because of the "mass constraint" (the 500g rule). They invented a "fake" gradient that always points the right way, allowing them to navigate the tricky terrain.

4. Why Should You Care?

You might ask, "Who cares about math cakes?"

Physics: These equations describe real-world phenomena like Bose-Einstein condensates (super-cold atoms) and laser beams. Knowing that there are "monster" solutions (high energy, sign-changing) helps physicists understand what happens when these systems get unstable or chaotic.
Mathematics: This paper solves a long-standing puzzle. Before this, mathematicians knew positive solutions existed, but finding many sign-changing solutions with specific properties was a huge open problem. They didn't just find one; they found a whole family of them and described exactly how they behave as the mass changes.

Summary

Think of the authors as explorers mapping a strange, invisible landscape.

They mapped the "Positive" islands and found there are exactly two of them in the "Explosive" zone.
Using that map, they built a bridge to the "Sign-Changing" continent.
They discovered that in this continent, there are infinitely many wild, complex shapes (solutions) that get more extreme as the "mass" gets smaller.

They proved that even with a strict rule (fixed mass), nature (or math) is incredibly diverse, offering infinite possibilities for how things can arrange themselves.

1. Problem Statement

The authors investigate the nonlinear Schrödinger (NLS) equation on a bounded domain $\Omega \subset \mathbb{R}^N$ with a prescribed $L^2$ -mass (norm) constraint. The problem is formulated as finding pairs $(u, \lambda) \in H^1_0(\Omega) \times \mathbb{R}$ satisfying:
$\begin{cases} -\Delta u + \lambda u = |u|^{p-2}u & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \\ \int_{\Omega} |u|^2 dx = \mu, \end{cases}$
where $\mu > 0$ is a fixed constant (the mass), and $\lambda$ is an unknown Lagrange multiplier. The exponent $p$ satisfies $2 < p < 2^*$ (Sobolev subcritical), where $2^* = \frac{2N}{N-2}$ for $N \ge 3$ and $2^* = \infty$ for $N=1,2$ .

The study focuses on three regimes based on the $L^2$ -critical exponent $p_c = 2 + \frac{4}{N}$ :

$L^2$ -subcritical: $2 < p < p_c$ .
$L^2$ -critical: $p = p_c$ .
$L^2$ -supercritical: $p_c < p < 2^*$ .

The primary objectives are:

To determine the exact number of positive solutions for small mass $\mu$ in the supercritical case on the unit ball $B_1$ .
To establish the existence and multiplicity of sign-changing solutions (nodal solutions) with prescribed mass across all regimes.
To analyze the asymptotic behavior of the energy $E(u)$ and the Lagrange multiplier $\lambda$ as $\mu \to 0^+$ and $\mu \to +\infty$ .

2. Methodology

The authors employ a sophisticated combination of variational methods, topological tools, and blow-up analysis. Key methodological components include:

Genus Theory: Used to establish the multiplicity of sign-changing solutions. By constructing symmetric sets with high genus on the $L^2$ -sphere $S_\mu(\Omega)$ , they define minimax levels to find critical points.
Descending Flow Techniques & Flow Invariance: A crucial innovation in this work. The authors develop a pseudo-gradient flow on the manifold $S_\mu(\Omega)$ that preserves the invariance of specific subsets (cones of positive/negative functions). This allows them to prove that critical points obtained via minimax arguments are indeed sign-changing (i.e., they do not converge to positive solutions).
Approximating Problems & Monotonicity Trick: To handle the lack of compactness in the supercritical case, they introduce a parameter $\tau \in [1/2, 1]$ in the nonlinearity ( $|u|^{p-2}u \to \tau|u|^{p-2}u$ ). They use Jeanjean's monotonicity trick to find bounded Palais-Smale sequences for almost every $\tau$ .
Morse Index Analysis: They utilize the boundedness of the Morse index of solutions to control the asymptotic behavior of the Lagrange multiplier $\lambda$ and the energy $E(u)$ via blow-up analysis.
Saddle-Point Geometry: For the supercritical case, they generalize the Mountain Pass geometry to a saddle-point structure to find solutions with large energy.
Precise Analysis on the Unit Ball: For the exact count of positive solutions, they perform a detailed analysis of the curve of positive solutions parameterized by $\lambda$ on the unit ball $B_1$ , utilizing scaling arguments and uniqueness results.

3. Key Contributions and Results

A. Multiplicity of Sign-Changing Solutions (Theorem 1.1)

The paper provides a comprehensive existence theory for sign-changing solutions with prescribed mass:

Subcritical Case ( $2 < p < p_c$ ): For any $\mu > 0$ , there exist infinitely many sign-changing solutions. Their energies tend to $+\infty$ as the index increases. As $\mu \to 0^+$ , the energy of the $j$ -th solution tends to 0, and $-\lambda$ tends to the $j$ -th Dirichlet eigenvalue of $-\Delta$ . As $\mu \to +\infty$ , energy tends to $-\infty$ and $\lambda \to +\infty$ .
Critical Case ( $p = p_c$ ): For sufficiently small $\mu > 0$ , there exist infinitely many sign-changing solutions with similar asymptotic behaviors to the subcritical case as $\mu \to 0^+$ .
Supercritical Case ( $p_c < p < 2^*$ ): For any integer $j$ , there exists a threshold $\mu^*_{p,j}$ such that for $0 < \mu < \mu^*_{p,j}$ , there are at least $j$ distinct sign-changing solutions. As $\mu \to 0^+$ , their energies tend to 0 and $-\lambda$ tends to Dirichlet eigenvalues.

Significance: This is the first result establishing the existence of infinitely many sign-changing solutions with prescribed mass on bounded domains. Previous works often found only one or required the mass to be in a range where no positive solutions exist.

B. Exact Number of Positive Solutions (Theorem 1.2)

Focusing on the unit ball $B_1$ in the supercritical regime ( $p_c < p < 2^*$ ):

For sufficiently small $\mu > 0$ , the problem admits exactly two positive solutions, denoted $u_{\mu, \tau}$ and $\tilde{u}_{\mu, \tau}$ .
Solution 1 (Low Energy): $E(u_{\mu, \tau}) \to 0$ and $\lambda_{\mu, \tau} \to -\lambda_1(B_1)$ as $\mu \to 0^+$ .
Solution 2 (High Energy): $E(\tilde{u}_{\mu, \tau}) \to +\infty$ and $\tilde{\lambda}_{\mu, \tau} \to +\infty$ as $\mu \to 0^+$ .
These results hold uniformly with respect to the parameter $\tau \in [1/2, 1]$ .

C. Existence of High-Energy Sign-Changing Solutions (Theorem 1.4)

In the supercritical case on the unit ball $B_1$ :

There exists a sign-changing solution $u_{sc, \mu}$ for small $\mu$ such that both its energy and Lagrange multiplier tend to $+\infty$ as $\mu \to 0^+$ .
This result relies on the exact count of positive solutions (Theorem 1.2) to distinguish the sign-changing solution from the positive ones using the descending flow technique.

4. Technical Innovations

Abstract Tool for Localization: The authors construct an abstract framework to localize bounded Palais-Smale sequences with Morse index information, ensuring the solutions found are sign-changing. This overcomes the difficulty that standard minimax methods often yield positive solutions or solutions that lose sign information in the limit.
Pseudo-Gradient Flow: They define a specific pseudo-gradient operator $V = \text{Id} - G$ (where $G$ solves a linear elliptic equation with a specific Lagrange multiplier) to construct a descending flow. This flow preserves the invariance of the set of sign-changing functions ( $S^*(\delta)$ ), a property not guaranteed by the standard gradient flow when $\lambda$ can be negative.
Uniformity in $\tau$ : The analysis of the exact number of positive solutions is shown to be independent of the parameter $\tau$ in the nonlinearity, which is crucial for the limiting argument used to find the high-energy sign-changing solution.

5. Significance

Completeness of Theory: The paper provides a complete picture of the solution structure (positive and sign-changing) for the NLS equation with prescribed mass on bounded domains across all subcritical, critical, and supercritical regimes.
Novelty in Multiplicity: It establishes the existence of infinitely many sign-changing solutions with prescribed mass, a significant advancement over previous literature which was limited to free mass problems or single sign-changing solutions.
Methodological Advancement: The development of the descending flow technique on manifolds with $L^2$ constraints and the precise control of the Lagrange multiplier's sign offer new tools for studying constrained variational problems where standard compactness fails.
Asymptotic Precision: The paper gives precise descriptions of the asymptotic behavior of solutions as the mass vanishes or diverges, linking the Lagrange multipliers directly to the spectrum of the Laplacian.

In summary, this work resolves long-standing questions regarding the multiplicity and asymptotic behavior of normalized solutions to the NLS equation on bounded domains, introducing powerful new variational techniques to handle sign-changing solutions in the presence of mass constraints.