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Lower order term for the fractional Laplacian with a Hardy potential

This paper establishes the existence and regularity of solutions for a semilinear elliptic equation involving the fractional Laplacian and a Hardy potential in a bounded domain, valid for any real parameter $\lambda$ under specific integrability conditions on the lower-order term.

Original authors: Rubén Fiñana, Alexis Molino

Published 2026-08-28

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Rubén Fiñana, Alexis Molino

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

In the vast landscape of mathematical physics, researchers often study how things change and settle into place. Imagine a stretched membrane, like a drumhead, but one that behaves in a strange, non-local way: when you push down on one spot, the entire surface feels the effect instantly, not just the immediate neighbors. This is the world of the fractional Laplacian, a tool used to describe systems where distant parts influence each other, such as in the movement of particles in a magnetic field or the stability of certain types of matter. Within this world, scientists also look at how forces that grow infinitely strong near a single point—like a singularity at the center of a domain—affect the system. This is known as a Hardy potential. For decades, mathematicians have tried to predict whether these complex systems will settle into a stable state or fly apart, especially when other forces are added to the mix.

A team of researchers has now taken a significant step forward in understanding these systems by adding a new, stabilizing element to the mix. They investigated a specific equation that models a system with three competing forces: the strange, long-range influence of the fractional Laplacian, the intense pull of the Hardy potential near the center, and a new, lower-order term that acts like a dampening agent. This dampening term is crucial because it represents a force that grows stronger as the system's activity increases, effectively acting as a brake. The researchers wanted to know if, despite the chaotic pull of the singularity and the complex long-range interactions, this braking force could guarantee that the system would always find a stable solution, regardless of how strong the central pull was.

The team proved that the answer is yes. They demonstrated that for a wide range of conditions, a stable solution always exists, no matter how strong the central attractive force becomes. In previous studies, researchers had found that solutions only existed if the central force stayed below a certain critical limit. However, by including this specific dampening term, the new work shows that the system can withstand any level of intensity from the central force. The researchers also showed that these solutions are not just theoretical possibilities but possess a high degree of smoothness and regularity, meaning they are well-behaved and predictable. This holds true even when the external inputs driving the system are quite rough or irregular.

To reach this conclusion, the authors had to navigate a mathematical landscape where standard tools often fail. They developed a method to handle the fact that the equations involved functions that could behave wildly near the center of the domain. By carefully analyzing how the different parts of the equation interact, they established that the stabilizing term is powerful enough to counteract the destabilizing effects of the singularity. Their work improves upon earlier findings that were limited to specific cases or required the central force to be weak. Now, it is known that as long as the dampening term is present and satisfies a mild condition regarding how it spreads across the domain, a solution is guaranteed to exist for any strength of the central force.

The implications of this finding extend to a variety of specific scenarios. The researchers showed that their results apply even when the stabilizing force itself changes depending on location, provided it remains positive. They also explored what happens when the external driving force is very rough, proving that the system still settles into a smooth, regular state. This is a significant improvement over previous knowledge, which suggested that the range of possible solutions would shrink or disappear entirely if the central force became too strong or if the external inputs were too chaotic. The new findings confirm that the presence of this lower-order term fundamentally changes the behavior of the system, ensuring stability where it was previously thought to be impossible.

In essence, this work provides a robust mathematical foundation for understanding complex systems where long-range interactions and intense local forces compete. It reassures scientists that by including a specific type of stabilizing mechanism, they can predict the behavior of these systems with confidence, even under extreme conditions. The results do not just fill a gap in the literature; they expand the boundaries of what is considered possible in the study of these non-local equations, offering a clearer picture of how order can emerge from complexity.

Technical Summary: Lower Order Term for the Fractional Laplacian with a Hardy Potential

Problem Statement
This paper investigates the existence and regularity of solutions to a semilinear elliptic equation involving the fractional Laplacian, a Hardy potential, and a lower-order term with a weight function. The problem is formulated in a bounded domain $\Omega \subset \mathbb{R}^N$ ( $N > 2s$ ) with $0 \in \Omega$ and $s \in (0, 1)$ :

$\begin{cases} (-\Delta)^s u + g(x)|u|^{p-1}u = \lambda \frac{u}{|x|^{2s}} + f(x), & \text{in } \Omega, \\ u = 0, & \text{in } \mathbb{R}^N \setminus \Omega, \end{cases}$

where $p > 1$ , $\lambda \in \mathbb{R}$ , $g \in L^1_{\text{loc}}(\Omega)$ is a positive function, and the source term $f$ belongs to the weighted Lebesgue space $L^{(p+1)/p}_g(\Omega)$ . The study aims to generalize previous results on the fractional Hardy equation (where $g \equiv 0$ ) by incorporating the lower-order term $g(x)|u|^{p-1}u$ under minimal integrability assumptions on $g$ .

Methodology and Functional Setting
The authors employ variational methods within the framework of fractional Sobolev spaces. The natural energy space for the problem is defined as $E = H^s_0(\Omega) \cap L^{p+1}_g(\Omega)$ , equipped with the norm $\|u\|_E = \|u\|_{H^s_0(\Omega)} + |u|_{L^{p+1}_g(\Omega)}$ .

The solutions are characterized as critical points of the energy functional $\phi_\lambda: E \to \mathbb{R}$ :
$\phi_\lambda(u) = \frac{1}{2}\|u\|^2_{H^s_0(\Omega)} + \frac{1}{p+1}\int_\Omega |u|^{p+1}g \, dx - \frac{\lambda}{2}\int_\Omega \frac{u^2}{|x|^{2s}} \, dx - \int_\Omega f u g \, dx.$

Key analytical tools utilized include:

Hardy Inequality: The fractional Hardy inequality is central to controlling the singular potential term $\lambda u/|x|^{2s}$ .
Ekeland's Variational Principle: Used to establish the existence of a minimizing sequence for $\phi_\lambda$ even in the absence of lower semicontinuity.
Compactness and Convergence: The proof of existence relies on showing that the minimizing sequence is bounded in $E$ , converges weakly, and that the nonlinear terms converge appropriately using Vitali's Theorem and the Dominated Convergence Theorem.
Test Function Techniques: For regularity results, the authors utilize truncated test functions of the form $|T_k(u)|^\rho T_k(u)$ , where $T_k$ is the standard truncation operator, to derive a priori estimates on the summability of the solution.

Key Contributions and Results

Existence of Solutions for All $\lambda \in \mathbb{R}$ :
The paper proves that a solution $u \in E$ exists for every real $\lambda$ , provided $g$ satisfies a specific mild integrability condition:
$\frac{1}{|x|^{2s}} g^{\frac{2}{p+1}} \in L^{\frac{p+1}{p-1}}(\Omega). \quad (\text{Condition 5})$
This result extends previous work (e.g., Abdellaoui et al. [1] for $g \equiv 0$ and Mi et al. [10] for $g \equiv 1$ ) by removing the restriction that $\lambda$ must be positive or bounded by the Hardy constant $\Lambda_{N,s}$ . The existence is guaranteed regardless of the sign of $\lambda$ .
Regularity and Improved Summability:
Under the assumption that $f \in L^q_g(\Omega)$ for $q \ge (p+1)/p$ and an additional condition on $g$ :
$\int_\Omega |x|^{\frac{2spq}{1-p}} g^{\frac{1-pq}{p-1}} \, dx < +\infty, \quad (\text{Condition 6})$
the authors prove that the solution $u$ possesses higher regularity, specifically $u \in H^s_0(\Omega) \cap L^{pq}_g(\Omega)$ . This demonstrates that the lower-order term enhances the summability of the solution compared to the case without the term.
Minimality of Solutions:
The paper establishes that if a stronger integrability condition on $g$ is met (Condition 14), the solution obtained is not merely a critical point but a global minimum of the functional $\phi_\lambda$ .

Significance and Claims
The authors claim that their results represent a significant improvement over existing literature in two main aspects:

Relaxation of Constraints on $\lambda$ : Unlike previous studies (such as Mi et al. [10]) where existence was often restricted to $0 < \lambda \le \Lambda_{N,s}$ , this work guarantees existence for every $\lambda \in \mathbb{R}$ .
Enhanced Regularity: For the case $g \equiv 1$ , the paper establishes solutions in $H^s_0(\Omega) \cap L^{pq}(\Omega)$ for a broader range of parameters ( $p > 2^*_s - 1$ ) compared to [10, Theorem 1.5]. The authors note that in [10], the interval of admissible $\lambda$ converges to zero as $p \to +\infty$ , whereas their results hold for all $\lambda$ .
Generality: The low regularity requirements on $g$ (only $L^1_{\text{loc}}$ and satisfying the integral conditions) allow the extension of these results to a wider class of problems, including those with multiple Hardy-type potentials (as illustrated in Corollary 4.4).

The paper concludes that the inclusion of the lower-order term, under the derived integrability conditions, effectively stabilizes the problem, allowing for existence and regularity results that are robust with respect to the parameter $\lambda$ and the summability of the source term $f$ .

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