Strong maximum principle for fully nonlinear nonlocal problems

1. Problem Statement

The paper investigates the solvability and qualitative properties of nonnegative solutions to a sublinear, fully nonlinear, nonlocal integro-differential equation of the form:
$$\begin{cases} \mathcal{M}^{\pm}[u] + a(x)u^{q(x)} = 0 & \text{in } \Omega, \\ u \geq 0 & \text{in } \Omega, \end{cases}$$
where:

• $\Omega \subset \mathbb{R}^n$ is a bounded $C^{1,1}$ convex domain.
• $\mathcal{M}^{\pm}$ are the Pucci extremal operators defined via a class $\mathcal{L}^*$ of homogeneous kernels (generalizing fractional Laplacians).
• $q \in (0, 1)$ represents the sublinear exponent.
• $a \in C(\Omega)$ is a weight function that may change sign.
• The solution $u$ is defined on the whole space $\mathbb{R}^n$, but the equation holds only inside $\Omega$.

Core Objective: The authors aim to determine conditions under which the Strong Maximum Principle (SMP) holds. Specifically, they investigate whether nontrivial solutions must be strictly positive in $\Omega$ or if they can exhibit "dead cores" (regions where $u \equiv 0$ within $\Omega$). The study focuses on how the negative part of the solution outside the domain ($u^-$ in $\mathbb{R}^n \setminus \Omega$) and the negative part of the potential ($a^-$) influence the formation of these dead cores.

2. Methodology

The authors employ a combination of viscosity solution theory, barrier methods, and Liouville-type theorems.

• Viscosity Solutions: The analysis is conducted within the framework of viscosity solutions, utilizing the definitions of subsolutions and supersolutions adapted for nonlocal operators.
• Barrier Construction: A crucial step involves constructing specific barrier functions (Lemma 2) involving radial functions $\rho_1$ and $\rho_2$ to handle the singular behavior near the boundary and the influence of the potential $V$.
• Hopf Lemma for Nonlocal Operators: The authors prove a new Hopf Lemma (Theorem 3) for fully nonlinear nonlocal operators. This lemma establishes a lower bound on the growth of a nonnegative solution near the boundary point where it vanishes, specifically showing that $u(x) \sim |x-x_0|^s$ near the boundary. This result extends previous work (e.g., [9]) to the fully nonlinear setting with potential terms.
• Liouville Theorems and Compactness: To prove the nonexistence of dead cores, the authors use a blow-up argument. They assume a sequence of solutions with vanishing norms or specific properties and rescale them. By establishing uniform lower bounds on the maximum of nontrivial solutions (via Liouville theorems), they show that the limit cannot be a nontrivial solution to the limiting equation (which admits no positive supersolutions in the sublinear case).
• Comparison Principles: The proofs rely heavily on comparison principles between the solution and the constructed barriers, taking into account the nonlocal nature of the operator (integrals over $\mathbb{R}^n$).

3. Key Contributions

A. A New Nonlocal Hypothesis on the Exterior Data

The paper introduces a novel condition regarding the negative part of the solution outside the domain ($u^-$).

• In local PDEs, the sign of the solution outside the domain is irrelevant. In nonlocal problems, the value of $u$ in $\mathbb{R}^n \setminus \Omega$ directly influences the operator $\mathcal{M}[u]$ inside $\Omega$.
• The authors demonstrate that if the $L^1_s$-norm of the negative part of the solution outside the domain is small, the SMP holds. If this norm is large, dead cores can form.

B. Proof of a Strong Maximum Principle (SMP)

The authors establish sufficient conditions for the SMP to hold, ensuring that any nontrivial solution is strictly positive in $\Omega$.

• Theorem 1: If $a^+ \not\equiv 0$ and the $L^1_s$-norm of $u^-$ (outside $\Omega$) is sufficiently small ($\|u^-\|_{L^1_s} < \delta$), then every nontrivial solution belongs to the set $P_0(\Omega)$ (solutions that are positive in $\Omega$ and have a negative normal derivative on the boundary).
• Theorem 2: If the weight $a(x)$ is strictly positive in $\Omega$, the sign condition on the exterior is not strictly necessary for existence, though a smallness condition on $\|u^-\|$ is still required to guarantee the SMP.

C. A New Hopf Lemma

The paper provides a rigorous proof of a Hopf Lemma for viscosity solutions driven by fully nonlinear nonlocal operators (Theorem 3). This lemma is of independent interest and is essential for proving the boundary behavior of solutions, showing that solutions cannot vanish "flatly" at the boundary unless they are identically zero.

D. Existence and Non-Existence Results

• Existence (Theorem 4): The authors prove the existence of nontrivial solutions for the Dirichlet problem with exterior data $g$, provided the negative part of $g$ is small (or $a$ is strictly positive).
• Violation of SMP (Example 1): The paper constructs a specific example where the SMP fails. By taking a sequence of solutions with increasingly large negative exterior data ($\|u^-\|_{L^1_s} \to \infty$), they show that the solution can be distorted until it touches zero inside the domain, creating a dead core. This demonstrates that the smallness condition on $u^-$ is necessary.

4. Main Results Summary

1. Non-existence of Dead Cores (Small Exterior Data):
   If the negative part of the solution outside the domain is small in the weighted $L^1_s$ norm, and the positive part of the weight $a$ is non-trivial, then any nontrivial solution $u$ satisfies $u > 0$ in $\Omega$. The solution behaves like $d(x)^s$ near the boundary (where $d(x)$ is the distance to the boundary).

2. Necessity of the Smallness Condition:
   If the negative part of the exterior data is large, the Strong Maximum Principle can be violated. The paper provides a constructive example where a solution exists that is non-negative in $\Omega$ but vanishes on a non-empty subset (a dead core), despite $a(x) > 0$.

3. Regularity:
   The paper derives regularity results for solutions, showing that near the boundary of the positivity set $\{u > 0\}$, solutions satisfy $|u(y)| \leq C|y-x|^{2s}$ under specific conditions, and $u \in C^{1,\alpha}$ locally.

5. Significance

• Theoretical Advancement: This work bridges a gap in the theory of nonlocal equations by addressing the Strong Maximum Principle in the context of fully nonlinear operators (Pucci operators) rather than just linear fractional Laplacians.
• Understanding Nonlocality: It highlights a fundamental difference between local and nonlocal problems: the exterior data is not just a boundary condition but an active component of the equation's dynamics. The "negative part" of the exterior data acts as a source of "dead cores," a phenomenon impossible in local elliptic equations with the same structure.
• Applications: The results are relevant for modeling physical and biological processes (e.g., combustion, chemical reactions) where sublinear reaction terms and nonlocal interactions (long-range diffusion) are present. The ability to predict when a reaction zone (dead core) will form based on exterior conditions is crucial for such applications.
• Methodological Tool: The new Hopf Lemma and the specific barrier constructions developed here provide tools that can be applied to other fully nonlinear nonlocal problems.

In conclusion, the paper rigorously establishes that for sublinear fully nonlinear nonlocal problems, the Strong Maximum Principle holds if and only if the negative influence from the exterior domain is sufficiently small. This provides a sharp dichotomy between solutions that are strictly positive and those that develop dead cores.