Liouville type theorems for fully nonlinear elliptic equations with superlinear growth in gradient

Technical Summary: Liouville Type Theorems for Fully Nonlinear Elliptic Equations with Superlinear Growth in Gradient

Problem Statement
This article investigates the nonexistence of positive supersolutions for a system of fully nonlinear elliptic equations driven by the Pucci extremal operators in exterior domains ($\mathbb{R}^n \setminus B_{R_0}$). The system is given by:
$$\begin{cases} -M^+_{\lambda,\Lambda}(D^2u) + |\nabla u|^q \ge \lambda_1 f_1(v) & \text{in } \mathbb{R}^n \setminus B_{R_0}, \\ -M^+_{\lambda,\Lambda}(D^2v) + |\nabla v|^q \ge \lambda_2 f_2(u) & \text{in } \mathbb{R}^n \setminus B_{R_0}, \end{cases}$$
where $n \ge 2$, $q > 1$, $\lambda_1, \lambda_2 > 0$, and $f_1, f_2$ are nondecreasing functions exhibiting power-type behavior near the origin or at infinity. The study focuses on solutions within the Sobolev regularity class $W^{2,p}_{loc}$ ($p > n$), rather than the classical $C^2$ framework, which is the natural regularity for fully nonlinear uniformly elliptic equations.

Methodology
The authors develop a framework to overcome the lack of pointwise second-order differentiability and the absence of variational structures or integral identities typical of fully nonlinear equations. The methodology relies on several key technical tools:

1. Radial Reduction and Classification: Using comparison principles and the rotational invariance of the Pucci operator, the authors establish that the existence of a general positive supersolution implies the existence of a positive radially symmetric supersolution. They classify these radial solutions into two categories: those that decay to zero at infinity and those that exhibit blow-up behavior ($u, v \to +\infty$).
2. Effective Dimension: A central concept introduced is the "effective dimension" $e_n^+ = \frac{\lambda}{\Lambda}(n-1) + 1$. This quantity replaces the Euclidean dimension $n$ in the qualitative analysis of solutions involving Pucci operators.
3. Nonlinear Transformations and Monotonicity: To handle the $W^{2,p}$ regularity, the authors employ nonlinear transformations adapted to the gradient structure. They utilize level-crossing principles for absolutely continuous radial profiles and transversality properties of auxiliary quantities to establish eventual monotonicity and convexity of the solutions.
4. Gradient Domination and Doubling Arguments: The analysis involves deriving sharp lower and upper decay estimates. For the case $q > q_c$, the authors adapt the "doubling lemma" (introduced by Birindelli and Demengel) to the Pucci setting, performing rescaling arguments to derive limiting equations and identify contradictions.
5. Critical Exponents: The study identifies a critical exponent $q_c = \frac{e_n^+}{e_n^+ - 1}$ which governs the qualitative behavior of the supersolutions. The analysis is divided into three regimes: $q < q_c$, $q = q_c$, and $q > q_c$.

Key Contributions and Results

• Sharp Liouville-Type Theorems: The paper establishes optimal nonexistence criteria for positive supersolutions in exterior domains. The results are sharp, meaning that if the conditions for nonexistence are violated, explicit positive supersolutions can be constructed.
• Regularity Level: Unlike previous works on similar systems (e.g., Burgos-Pérez and García-Melián) which relied on $C^2$ solutions, this work establishes the theory at the $W^{2,p}_{loc}$ level ($p>n$), which is the standard regularity for fully nonlinear equations.
• Role of Effective Dimension: The authors demonstrate that the critical thresholds for nonexistence are determined by the interaction between the gradient exponent $q$, the effective dimension $e_n^+$, and the asymptotic behavior of the nonlinearities $f_1, f_2$.
• Regime Analysis:
  • Case $1 < q < q_c$: Nonexistence is proven when $p_1 p_2 \le q^2$ or when specific inequalities involving the exponents $\alpha_1, \alpha_2$ and the critical value $\beta = \frac{2-q}{q-1}$ are satisfied.
  • Case $q = q_c$: Nonexistence results are derived for $p_1 p_2 \le q_c^2$ and specific ranges of $p_1 p_2 > q_c^2$.
  • Case $q > q_c$: The analysis becomes more complex, involving two pairs of critical exponents ($\bar{\alpha}_i$ and $\hat{\alpha}_i$). The paper provides nonexistence conditions based on the interplay between these exponents and the effective dimension.
  • Blow-up Behavior: The paper also addresses supersolutions that blow up at infinity, proving their nonexistence under certain conditions on the nonlinearities at infinity (Theorem 2.5).
• Optimality: The authors provide counterexamples (constructing explicit radial supersolutions) to show that their nonexistence conditions are optimal. These constructions involve balancing the decay rates of the Pucci terms, gradient terms, and reaction terms.

Significance
The paper claims to fill a significant gap in the literature by extending the Liouville theory for semilinear elliptic systems with nonlinear gradient terms to the fully nonlinear setting driven by Pucci extremal operators. It reveals the fundamental role of the effective dimension $e_n^+$ in determining critical nonexistence thresholds, analogous to how the Euclidean dimension functions in the Laplacian case. By working within the $W^{2,p}$ framework, the results provide a more robust and natural theory for fully nonlinear equations, avoiding the artificial assumption of classical $C^2$ regularity. The findings serve as a fully nonlinear analogue to the Liouville theory for semilinear systems and provide a framework for studying similar problems beyond the classical setting.