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Double phase inequalities with convolution nonlinearity in exterior domains

This paper establishes the existence of $C^1$ -solutions for double phase inequalities involving convolution nonlinearity in exterior domains, revealing a sharp distinction where the inequality with a positive sign admits solutions under broad conditions while the negative sign requires specific constraints on the exponents, with solvability linked to corresponding equations via a novel sub- and supersolution method.

Original authors: Marius Ghergu

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Marius Ghergu

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 standing in an infinite, open field (mathematicians call this an "exterior domain") outside a small, forbidden circle in the center. Your goal is to find a "shape" or a "hill" (a mathematical function called $u$ ) that sits in this field and obeys a very specific set of physical rules.

This paper is about figuring out when such a shape can actually exist. The rules are complicated because they involve two different types of "terrain physics" fighting against each other, plus a weird, long-distance influence.

Here is the breakdown of the paper's story, using simple analogies:

1. The Two Types of Terrain (The Operators)

The paper studies a shape defined by two competing forces, represented by two mathematical operators:

The $s$ -Laplace Operator ( $\Delta_s$ ): Think of this as a standard, smooth rubber sheet. It wants to flatten out. It represents a "standard" way things spread or smooth out in space.
The $L_g$ Operator: This is the "double phase" part. Imagine a rubber sheet that changes its stiffness depending on how much you stretch it. In some places, it acts like a stiff metal rod; in others, it acts like a flexible rope. The paper specifically looks at cases where this operator is stiffer than the standard rubber sheet.

The paper asks: If you push these two forces together, can you create a stable hill?

2. The Long-Distance Whisper (The Convolution)

The most unique part of this problem is the term $(|x|^{-\alpha} * u^p)u^q$ .

The Analogy: Imagine that every point on your hill can "hear" every other point on the hill, but the sound gets quieter the farther away it is.
The Math: The value of the hill at one spot isn't just determined by its immediate neighbors; it's determined by a weighted average of the entire hill across the whole infinite field. This is called a "convolution." It's like a global feedback loop where the shape of the whole mountain influences the shape of a single rock.

3. The Two Scenarios: Pushing vs. Pulling

The paper investigates two different equations, which act like two different weather patterns:

Scenario A: The "Plus" Case ( $P+$ )

The Equation: $L_g + \Delta_s \ge \text{The Long-Distance Whisper}$
The Analogy: Imagine you are trying to build a hill where the ground helps you. The two terrain forces are working together (or at least not fighting too hard) to support the shape against the long-distance whisper.
The Result: The paper proves that you can almost always build this hill. No matter how you tune the stiffness of the ground or the height of the whisper, as long as the basic rules of geometry are met, a solution exists. It's like saying, "If you have enough help, you can always build a sandcastle."

Scenario B: The "Minus" Case ( $P-$ )

The Equation: $L_g - \Delta_s \ge \text{The Long-Distance Whisper}$
The Analogy: Now, the two terrain forces are fighting each other. One wants to flatten the hill, the other wants to keep it steep. They are in a tug-of-war, and the "Long-Distance Whisper" is trying to blow the hill away.
The Result: This is much harder. The paper finds a sharp line between success and failure.

If the "whisper" (the exponents $p$ and $q$ ) is too weak, or if the ground is too stiff in the wrong way, no hill can exist. The forces cancel each other out, or the whisper blows the shape away.
However, if the parameters are "sufficiently large" (meaning the whisper is strong enough and the math balances out just right), then a solution does exist.

4. The "Goldilocks" Conditions

For the difficult "Minus" case, the authors found the exact "Goldilocks" conditions (not too hot, not too cold) required for a solution to exist. They proved that a solution exists if and only if the numbers $p$ and $q$ (which control the strength of the long-distance whisper) satisfy specific inequalities involving the dimension of the space ( $N$ ) and the decay rate ( $\alpha$ ).

If these numbers are too small, the hill collapses. If they are just right, the hill stands.

5. How They Solved It

The authors didn't just guess; they built a mathematical "scaffold."

Sub and Supersolutions: They built a "ceiling" (a shape that is definitely too big) and a "floor" (a shape that is definitely too small).
The Sandwich: They showed that if the ceiling and floor are close enough and the rules are right, there must be a perfect shape squeezed right in the middle.
Integral Estimates: They used a method of measuring the "total volume" of the hill in different rings around the center to prove that the hill doesn't grow too fast or shrink too fast.

Summary

In plain English, this paper is a guidebook for a specific type of mathematical landscape.

Good News: If the forces are helping each other (the "Plus" case), a solution is guaranteed.
Bad News (with a catch): If the forces are fighting (the "Minus" case), a solution only exists if the "global influence" of the shape is strong enough to overcome the internal conflict. The authors mapped out the exact boundary between "impossible" and "possible."

The paper does not discuss real-world applications like engineering or medicine; it is purely a theoretical exploration of when these specific mathematical shapes can exist in an infinite space.

Technical Summary: Double Phase Inequalities with Convolution Nonlinearity in Exterior Domains

Problem Statement
This paper investigates the existence of $C^1$ -solutions for two related double phase elliptic inequalities defined in the exterior domain $\mathbb{R}^N \setminus B_1$ (where $N \ge 1$ ). The inequalities are given by:
$(P \pm) \quad L_g u \pm \Delta_s u \ge (|x|^{-\alpha} * u^p)u^q$
Here, $\Delta_s u = \text{div}(|\nabla u|^{s-2}\nabla u)$ is the $s$ -Laplace operator ( $s > 1$ ), and $L_g u = -\text{div}(|\nabla u|^{m-2}g(|\nabla u|)\nabla u)$ is a generalized operator with $m > s > 1$ . The function $g: [0, \infty) \to (0, \infty)$ is non-increasing, $C^1$ on $(0, \infty)$ , continuous at 0, and satisfies specific asymptotic conditions near the origin. The term $(|x|^{-\alpha} * u^p)$ represents a convolution over the exterior domain, introducing a nonlocal nonlinearity. The study covers cases where $L_g$ reduces to the $m$ -Laplace operator or the $m$ -mean curvature operator.

Methodology
The author employs a combination of integral estimates, comparison principles, and a novel sub- and supersolution method adapted for Orlicz spaces to handle the nonlocal convolution term in an unbounded domain.

Comparison Principles: A comparison principle is established for the operator $L_g - \Delta_s$ in bounded domains, relying on the monotonicity of the function $t \mapsto tA(t)$ where $A(t) = t^{m-2}g(t) + t^{s-2}$ .
Radial Estimates: For the existence of solutions to $(P+)$ , radially symmetric test functions of the form $u(x) = \kappa|x|^{-\gamma}$ are constructed. The decay rates are carefully chosen to balance the growth of the differential operators against the convolution term, utilizing Lemma 2.2 to estimate the convolution of power functions.
A Priori Estimates: For the non-existence and existence results regarding $(P-)$ , the author derives integral estimates on annuli $B_{3R} \setminus B_{2R}$ . By testing the inequality with specific cut-off functions and applying Young's and Hölder's inequalities, bounds on the integrals of powers of the solution are established.
Sub- and Supersolution Iteration: To prove the existence of solutions for $(P-)$ , a new iterative scheme is devised. A supersolution is constructed using a power law decay, while a subsolution is derived from the behavior of the operator at infinity. An increasing sequence of solutions is constructed on expanding bounded domains $B_n \setminus B_1$ , utilizing the compactness of the embedding in Orlicz-Sobolev spaces ( $W^{1,G}_{loc}$ ) to pass to the limit and obtain a global $C^1$ solution.

Key Contributions and Results

Distinct Behavior of $(P+)$ vs. $(P-)$ : The paper reveals a sharp dichotomy between the two inequalities.
- Theorem 1.1: For the inequality $(P+)$ (where the operators are added), positive solutions exist for all parameters $m > s > 1$ and $q > s-1$ , provided $g$ satisfies the basic regularity condition.
- Theorem 1.2: For the inequality $(P-)$ (where the operators are subtracted), no positive solutions exist if $m > s \ge N$ .
Optimal Conditions for $(P-)$ :
- Theorem 1.3: For $m > s > 1$ and $s < N$ , the existence of a positive solution to the inequality $(P-)$ is equivalent to the existence of a positive $C^1$ -solution to the corresponding equation $L_g u - \Delta_s u = (|x|^{-\alpha} * u^p)u^q$ .
- The paper derives necessary and sufficient conditions on the exponents $p$ and $q$ for solvability:
  $\min\{p, q\} > \frac{(s-1)(N-\alpha)}{N-s} \quad \text{and} \quad p + q > \frac{(s-1)(2N-\alpha)}{N-s}$
- These conditions are shown to be optimal. The proof involves ruling out equality cases by deriving logarithmic improvements in the decay of solutions, which lead to contradictions with the integral estimates.
Specific Operators: Corollary 1.4 applies these general results to the specific cases of the $m$ -Laplace operator ( $-\Delta_m u - \Delta_s u$ ) and the $m$ -mean curvature operator ( $-H_m u - \Delta_s u$ ), confirming that the same exponent conditions govern their solvability.

Significance
The paper claims to bridge the gap between double phase problems (relevant to heterogeneous structures in physics) and nonlocal PDEs featuring convolution terms (originally introduced in quantum physics models). The primary significance lies in:

Establishing a sharp distinction between the solvability of the sum and difference of double phase operators in exterior domains.
Providing optimal existence conditions for the difference case $(P-)$ , which is more restrictive and requires specific relationships between the dimension $N$ , the convolution parameter $\alpha$ , and the nonlinearity exponents $p, q$ .
Introducing a new sub- and supersolution method suitable for Orlicz spaces that effectively handles the nonlocal convolution term in unbounded domains, differing from previous approaches found in the literature.

The work does not propose new physical applications or experimental proposals but rather provides a rigorous mathematical framework for understanding the existence of solutions to these specific classes of quasilinear elliptic inequalities.