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Multiple positive solutions to a perturbed Gelfand problem involving mixed local-nonlocal operators and singular nonlinearity

This paper establishes the existence of multiple positive solutions, including a three-solution theorem, for a perturbed Gelfand problem involving a mixed local-nonlocal $p$ -Laplacian with singular nonlinearity by employing a novel sub-supersolution method that avoids conventional ODE techniques and by proving a new Hopf-type Strong Comparison Principle.

Original authors: Sarbani Pramanik

Published 2026-02-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sarbani Pramanik

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 bake the perfect loaf of bread in a very specific, oddly shaped kitchen. You have a recipe (a mathematical equation) that tells you how the dough should rise based on two competing forces:

The Local Force: This is like the heat from the oven directly touching the dough. It spreads smoothly and predictably, just like a standard oven heating a pan. In math, this is the "local" part of the equation.
The Non-Local Force: This is like a magical gust of wind that can instantly affect the dough from across the room, or even from the next house over, without touching it directly. It represents long-range interactions. In math, this is the "non-local" part.

The paper you are asking about is about solving a very tricky version of this baking problem. Here is the breakdown in simple terms:

The Problem: A "Singularity" in the Recipe

The recipe has a special ingredient called a singularity. Imagine that as the dough gets very thin (close to zero), the recipe demands an infinite amount of yeast to keep it rising. Mathematically, this is the term $\frac{1}{u^\beta}$ . It makes the equation extremely difficult to solve because standard tools break down when things get too small.

The author is asking: "If we mix these two forces (local heat and non-local wind) and add this tricky infinite-yeast ingredient, can we find a solution? And if so, how many different 'perfect loaves' (solutions) can we bake?"

The Main Discovery: Finding Three Solutions

Usually, when mathematicians solve these types of problems, they hope to find one solution. Sometimes, they find two.

This paper proves that under the right conditions (specifically, when the "activation energy" of the reaction is high enough), you don't just get one or two solutions. You can find three distinct, positive solutions.

Think of it like a thermostat with a very strange dial:

Solution 1 (The Low Branch): The dough rises just a tiny bit. It's stable, but small.
Solution 2 (The Middle Branch): The dough rises to a medium height.
Solution 3 (The High Branch): The dough rises explosively high.

The paper shows that for a specific range of settings on your "dial" (the parameter $\lambda$ ), all three of these states are mathematically possible at the same time. This creates an "S-shaped" curve, which is a classic sign of complex behavior in physics and chemistry (like how a fire might suddenly ignite or suddenly go out).

How They Did It: Building a "Safety Net"

To prove these three solutions exist, the author used a clever construction method called Sub- and Supersolutions.

Imagine you are trying to catch a ball (the solution) in a net.

The Subsolution: You build a floor underneath the ball. You prove the ball must be higher than this floor.
The Supersolution: You build a ceiling above the ball. You prove the ball must be lower than this ceiling.

If you can build a floor and a ceiling that trap the ball, you know a solution exists between them.

The Innovation:

Old Way: Previous mathematicians tried to build these nets using simple one-dimensional tools (like looking at a single line of dough) or by using complex maps (Green's functions) that only work for simple shapes.
This Paper's Way: The author built a new kind of net specifically designed for this "mixed" kitchen (local + non-local). They didn't rely on the old, rigid tools. Instead, they crafted custom "helper functions" that fit perfectly into the complex, multi-dimensional shape of the problem. This makes the method much more flexible and powerful.

The "Strong Comparison" Rule

To prove the third solution exists (the one in the middle), the author had to prove a new rule called the Hopf-type Strong Comparison Principle.

Think of this as a rule about how two different loaves of dough compare to each other.

If Loaf A is slightly smaller than Loaf B everywhere inside the kitchen, this rule proves that Loaf B isn't just slightly bigger; it is strictly bigger, and the difference is visible right at the edges of the kitchen.
The author proved this rule works even with the tricky "infinite yeast" ingredient and the mix of local/non-local forces. This was a major hurdle because no one had proven this specific rule for this specific type of mixed problem before.

What's Left Unsolved?

The paper successfully finds three solutions in two specific scenarios:

When the "infinite yeast" isn't actually there (the non-singular case).
When the local force is perfectly linear (like a standard spring).

However, the author admits that if you have the "infinite yeast" AND a complex, non-linear local force (the general case), they could only prove two solutions exist. Finding the third solution in that most difficult, general scenario remains an open mystery for future mathematicians.

Summary

In short, this paper is a mathematical tour de force that:

Mixes two different types of physical forces (local and non-local).
Adds a difficult, "infinite" ingredient to the mix.
Invents new tools to prove that, surprisingly, three different stable outcomes are possible, not just one or two.
Establishes a new rule for comparing these outcomes, which is a significant step forward for the field.

Technical Summary: Multiple Positive Solutions to a Perturbed Gelfand Problem Involving Mixed Local-Nonlocal Operators and Singular Nonlinearity

Problem Statement
This paper investigates the existence and multiplicity of positive solutions to a perturbed Gelfand-type boundary value problem driven by a mixed local-nonlocal $p$ -Laplacian operator with a singular nonlinearity. The problem is formulated as:
$\begin{cases} -\Delta_p u + (-\Delta_p)_s u = \lambda \frac{f(u)}{u^\beta} & \text{in } \Omega, \\ u > 0 & \text{in } \Omega, \\ u = 0 & \text{in } \mathbb{R}^N \setminus \Omega, \end{cases} \quad (P_\lambda)$
where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $1 < p < \infty$ , $s \in (0, 1)$ , $\lambda > 0$ is a parameter, and $0 \le \beta < 1$ . The nonlinearity $f: [0, \infty) \to (0, \infty)$ is a $C^1$ non-decreasing function with $f(0) > 0$ . The operator combines the local $p$ -Laplacian, $\Delta_p u = \text{div}(|\nabla u|^{p-2}\nabla u)$ , and the nonlocal fractional $p$ -Laplacian, defined via a principal value integral.

The study is motivated by combustion models in chemical kinetics, specifically the small fuel loss steady-state model where the reaction rate follows an Arrhenius law. The inclusion of the singular term $u^{-\beta}$ and the mixed operator structure introduces significant analytical challenges, particularly regarding the regularity of solutions and the application of comparison principles.

Methodology
The primary analytical tool employed is the method of sub- and supersolutions, adapted to handle the singular nature of the nonlinearity and the mixed operator structure.

Functional Framework: The authors work within the space $X_0^{1,p}(\Omega)$ , defined as the completion of $C_c^\infty(\Omega)$ under a norm combining the $W^{1,p}$ norm and the Gagliardo seminorm. This space naturally incorporates the Dirichlet boundary condition $u=0$ in $\mathbb{R}^N \setminus \Omega$ .
Construction of Sub- and Supersolutions:
- A first pair of strict sub- and supersolutions $(\phi_1, \psi_1)$ is constructed for any $\lambda > 0$ using a small multiple of the solution to a linear problem and a large multiple of a specific singular solution $\psi_0$ .
- A second pair $(\phi_2, \psi_2)$ is constructed under specific structural conditions on $f$ (denoted as (f1)–(f5)). This construction relies on auxiliary functions and cut-off techniques tailored to the mixed setting, avoiding reliance on ODE techniques or Green's function estimates common in purely local or purely fractional contexts.
Fixed Point Theorems:
- For the existence of two solutions, the authors utilize the monotone iteration method within ordered intervals defined by the sub- and supersolutions.
- For the existence of a third solution, the paper employs Amann's fixed point theorem. This requires establishing that the solution map is completely continuous and that the sub- and supersolution pairs satisfy specific ordering and disjointness properties.
Strong Comparison Principles: A critical component of the methodology, particularly for the third solution, is the derivation of a Hopf-type Strong Comparison Principle (SCP) for the linear mixed operator with singular nonlinearity. This involves proving strict inequality between solutions and their normal derivatives on the boundary.

Key Contributions and Results

Existence of Two Solutions (Theorem 1.1): Under assumptions (f1)–(f3), the problem admits at least one positive solution for every $\lambda > 0$ . If $f$ further satisfies conditions (f4) and (f5), there exists an interval $(\lambda_*, \lambda^*)$ of parameters for which the problem admits at least two distinct positive solutions.
Existence of Three Solutions (Theorem 1.2 & 1.4): The paper establishes a "three-solution theorem" in two specific regimes:
1. Non-singular case ( $\beta = 0, p > 1$ ): For $\lambda \in (\lambda_*, \lambda^*)$ , there exist three distinct positive solutions.
2. Linear singular case ( $p = 2, 0 < \beta < 1$ ): For $\lambda \in (\lambda_*, \lambda^*)$ , there exist three distinct positive solutions.
  The bifurcation diagram in these cases is shown to possess an S-like structure with two turning points, corresponding to ignition and extinction limits.
Hopf-type Strong Comparison Principle (Theorem 1.3): The authors prove a new Strong Comparison Principle for the linear mixed operator with singular nonlinearity. Specifically, if $u_1, u_2$ satisfy the equation with $f_1 \le f_2$ (and $f_1 \not\equiv f_2$ ), then $u_1 < u_2$ in $\Omega$ and $\frac{\partial u_2}{\partial \nu} < \frac{\partial u_1}{\partial \nu} < 0$ on $\partial \Omega$ . This result holds for the full range $s \in (0, 1)$ , improving upon previous results that were restricted to $s < 1/2$ .
Novel Construction Techniques: The construction of sub- and supersolutions avoids conventional ODE techniques and Green's function estimates, offering a framework more adaptable to nonlinear and nonlocal operators.

Significance and Limitations
The paper claims to provide the first multiplicity and three-solution results for mixed local-nonlocal problems with singular nonlinearities. A significant contribution is the development of the Hopf-type SCP for singular problems in the linear mixed setting, which is described as a crucial tool for deriving the third solution and holds broader mathematical significance.

The authors explicitly note a limitation: the existence of a third solution for the general nonlinear singular case ( $p \neq 2$ and $0 < \beta < 1$ ) remains an open question. The primary obstacle is the difficulty in proving the Hopf-type Strong Comparison Principle for nonlinear mixed operators. The paper concludes by stating that a forthcoming work will address the development of a Hopf-type SCP for nonlinear and nonhomogeneous mixed operators, which would allow the extension of the three-solution theorem to the general case.

The results are presented as rigorous mathematical proofs within the framework of nonlinear elliptic PDEs, with specific attention to the interplay between local diffusion, nonlocal interactions, and singular reaction terms.