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Existence of solutions for elliptic problems involving the $(1,q)$ -Laplacian operator and a discontinuous superlinear nonlinearity

This paper establishes the existence of nontrivial nonnegative solutions for quasilinear elliptic problems involving the $(1,q)$ -Laplacian operator and a discontinuous superlinear nonlinearity by employing an approximation method via $(p,q)$ -Laplacian problems as $p \to 1^+$ , while also analyzing the asymptotic convergence of these solutions to a limit problem without discontinuity.

Original authors: Marcos A. V. Costa, Olímpio H. Miyagaki, Marcos T. O. Pimenta

Published 2026-06-19

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CC0 1.0

Original authors: Marcos A. V. Costa, Olímpio H. Miyagaki, Marcos T. O. Pimenta

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 with a twist: the rules for how the bubble behaves change depending on how big it gets. This is essentially what the mathematicians in this paper are doing, but instead of soap bubbles, they are solving complex equations that describe physical forces in a specific region (like a room or a container).

Here is a breakdown of their work using simple analogies:

The Two Forces at Play

The problem involves two different "forces" trying to shape a solution (let's call the solution a "shape" or a "curve"):

The "Rigid" Force (The 1-Laplacian): Think of this as a force that wants the shape to be as straight and efficient as possible, like a taut string. It's very strict and creates sharp corners. Mathematically, this is tricky because it lives in a "rough" world where standard smoothness rules don't apply.
The "Smooth" Force (The q-Laplacian): Think of this as a force that wants the shape to be soft, rounded, and flexible, like a rubber band. This force is easier to handle mathematically because it keeps things smooth.

The authors are studying what happens when you mix these two forces together. The "Rigid" force is the difficult part because it makes the math break down in standard ways.

The "On/Off" Switch (The Discontinuity)

The equation also has a special switch called the Heaviside function. Imagine a light switch that is either completely OFF (0) or completely ON (1).

If your "shape" is below a certain height (let's call it the "beta" line), the switch is OFF, and a specific force is zero.
If your "shape" goes above that line, the switch flips ON, and a strong force kicks in.

The problem is that this switch doesn't dim gradually; it snaps instantly. This "snap" makes the math very hard to solve because the rules change abruptly.

The Strategy: The "Training Wheels" Approach

Because the "Rigid" force (1-Laplacian) is so difficult to handle directly, the authors use a clever trick called approximation.

Step 1: The Practice Run. Instead of tackling the super-rigid "1" force immediately, they start with a slightly softer version (a "p-Laplacian" where p is a number just slightly bigger than 1). Think of this as putting training wheels on a bike. The bike is still a bit wobbly, but it's manageable.
Step 2: Solving the Easier Problem. They solve the problem with these "training wheels" (the softer force) combined with the "Smooth" force and the "On/Off" switch. They prove that a solution exists for this easier version.
Step 3: Removing the Training Wheels. Once they have the solution for the softer version, they slowly make the force get "stiffer" and "stiffer" (letting p get closer and closer to 1).
The Result: They show that as the training wheels come off, the solution doesn't fall apart. Instead, it settles into a stable, non-zero shape that solves the original, super-rigid problem.

The Main Discoveries

1. A Solution Exists (Theorem 1)
The authors proved that even with the tricky "Rigid" force and the snapping "On/Off" switch, a valid solution exists.

The Shape: The solution isn't just a rough sketch; thanks to the "Smooth" force helping out, the final shape is actually quite well-behaved (mathematically, it belongs to a specific "Sobolev space," meaning it has enough smoothness to be useful).
It's Not Empty: They proved the solution isn't just "zero" (nothing happening). There is a real, non-zero shape that forms.

2. What Happens When the Switch Gets Smaller? (Theorem 2)
The "beta" line (the height where the switch flips) is a variable. The authors asked: "What happens if we lower that switch height all the way to zero?"

The Limit: As they lower the switch height (letting beta approach 0), the family of solutions they found doesn't disappear. Instead, they all converge to a single, final solution.
The Final State: This final solution solves a slightly different version of the problem where the "On/Off" switch is gone, replaced by a continuous force. Essentially, they showed that the messy, snapping switch problem smoothly transitions into a clean, continuous problem as the switch height vanishes.

Why This Matters (In Their Words)

The paper doesn't claim to fix bridges or cure diseases directly. Instead, it builds a mathematical bridge.

The "Rigid" force (1-Laplacian) is naturally linked to Functions of Bounded Variation, a concept used to describe things with sharp edges or jumps (like the edge of a shadow or a crack in a material).
By proving that solutions exist and behave well even with these sharp edges and snapping switches, the authors provide a rigorous foundation. They show that you can use standard "smooth" math tools to understand "rough" problems, provided you use the right approximation strategy.

In summary: The paper is a mathematical proof that says, "Even if you mix a very rigid force with a snapping switch, you can still find a stable, non-zero solution by starting with a softer version and slowly tightening the rules until you reach the real problem."

Technical Summary: Existence of Solutions for Elliptic Problems Involving the (1, q)-Laplacian and a Discontinuous Superlinear Nonlinearity

Problem Statement
The paper investigates a class of quasilinear elliptic problems defined on a bounded domain $\Omega \subset \mathbb{R}^N$ ( $N \geq 2$ ) with a Lipschitz boundary. The central equation is:
$\begin{cases} -\Delta_1 u - \Delta_q u = H(u - \beta) |u|^{s-2} u & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases} \quad (P)$
where:

$\Delta_1 u = \text{div}\left(\frac{\nabla u}{|\nabla u|}\right)$ is the 1-Laplacian operator.
$\Delta_q u = \text{div}(|\nabla u|^{q-2}\nabla u)$ is the $q$ -Laplacian operator, with $1 < q < s < q^*$ (where $q^*$ is the critical Sobolev exponent).
$H$ is the Heaviside function, introducing a discontinuity at $u = \beta$ ( $\beta > 0$ ).
The nonlinearity is superlinear ( $s > q$ ).

The primary mathematical difficulty arises from the 1-Laplacian, which naturally leads to the space of Functions of Bounded Variation ( $BV(\Omega)$ ), a space lacking reflexivity. This creates significant technical challenges regarding compactness and the definition of weak solutions. Additionally, the presence of the discontinuous Heaviside function prevents the use of standard differentiable variational methods.

Methodology
The authors employ a multi-step approximation strategy combined with nonsmooth critical point theory and the theory of divergence-measure fields:

Approximation by (p, q)-Laplacian: The problem is approached by considering a family of auxiliary problems $(P_p)$ involving the $(p, q)$ -Laplacian operator ( $-\Delta_p u - \Delta_q u$ ) where $1 < p < q$ . The limit $p \to 1^+$ recovers the original $(1, q)$ -Laplacian structure.
Nonsmooth Variational Framework: Since the energy functional associated with the discontinuous nonlinearity is not Fréchet differentiable, the authors utilize the critical point theory for locally Lipschitz functionals (Clarke's generalized gradient). They define a functional $J_{p,\beta}$ and apply a nonsmooth version of the Mountain Pass Theorem to establish the existence of critical points $u_{p,\beta}$ .
Uniform Estimates and Compactness:
- Boundedness: The authors prove that the sequence of solutions $(u_{p,\beta})$ is uniformly bounded in $W^{1,q}_0(\Omega)$ and $L^\infty(\Omega)$ (via Moser iteration).
- Convergence: As $p \to 1^+$ , the sequence converges to a limit function $u_\beta$ . The term involving the 1-Laplacian is handled by identifying the weak limit of the vector field $|\nabla u_{p,\beta}|^{p-2}\nabla u_{p,\beta}$ as a vector field $z_\beta \in L^\infty(\Omega, \mathbb{R}^N)$ with $\|z_\beta\|_\infty \leq 1$ .
Limit Analysis ( $\beta \to 0^+$ ): The paper further analyzes the asymptotic behavior of the solutions $u_\beta$ as the parameter $\beta$ tends to zero. This involves proving the convergence of the family $(u_\beta)$ to a solution $u_0$ of the continuous limit problem $(P_0)$ , where the Heaviside discontinuity is removed.

Key Contributions and Results

Definition of Weak Solution: The paper establishes a rigorous definition of a solution for Problem (P) within the context of $BV$ functions and divergence-measure fields. A function $u$ $u$ is a solution if there exist a vector field $z$ $z$ and a function $\rho$ $ρ$ such that:
- $-\text{div}(z) - \Delta_q u = \rho$ in the distributional sense.
- $z \cdot \nabla u = |\nabla u|$ almost everywhere (a crucial condition linking the vector field to the gradient).
- $\rho$ satisfies a multivalued inclusion corresponding to the Heaviside function at the discontinuity point $u=\beta$ .
Theorem 1 (Existence): For any $\beta > 0$ , Problem (P) admits at least one nonnegative, nontrivial solution $u_\beta \in W^{1,q}_0(\Omega)$ . Notably, despite the presence of the 1-Laplacian, the solution belongs to the Sobolev space $W^{1,q}_0(\Omega)$ due to the regularizing effect of the $q$ -Laplacian term.
Theorem 2 (Asymptotic Behavior): As $\beta \to 0^+$ , the family of solutions $(u_\beta)$ converges (up to a subsequence) to a nontrivial, nonnegative solution $u_0 \in W^{1,q}_0(\Omega)$ of the limit problem $(P_0)$ (where the RHS is $|u|^{s-2}u$ ). The convergence holds in $L^r(\Omega)$ for $1 \leq r < q^*$ and almost everywhere.
Non-triviality: The authors prove that the solutions are nontrivial by showing that the measure of the set where $u_\beta(x) > \beta$ is bounded away from zero for small $\beta$ , ensuring the solution does not vanish or concentrate entirely below the threshold.

Significance
The paper contributes to the development of the theory for $(1, q)$ -Laplacian operators, a setting that remains largely unexplored compared to standard $p$ -Laplacian problems. By combining approximation techniques with nonsmooth analysis and divergence-measure field theory, the authors successfully address the dual challenges of the non-reflexive nature of the 1-Laplacian and the discontinuity of the nonlinearity. The work extends previous results on 1-Laplacian problems (e.g., those by Pimenta et al. and Figueiredo et al.) to the mixed $(1, q)$ case, demonstrating that the regularizing effect of the $q$ -Laplacian allows for solutions in Sobolev spaces while maintaining the necessary structure to handle the singular 1-Laplacian term.