AI 中文总结
研究完全非线性椭圆系统正上解，引入有效维数确定临界指数\(q_{c}\)，通过梯度指数、有效维数和非线性耦合相互作用建立刘维尔型不存在性定理，揭示有效维数在确定临界不存在阈值中的作用。
AI 中文摘要
本文研究完全非线性椭圆系统\[ \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)+|\nabla u|^{q} \geq\lambda_{1}f_{1}(v)~~\text{在}~~\mathbb{R}^{n}\setminus B_{R_0}中,\\ -\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}v)+|\nabla v|^{q} \geq\lambda_{2}f_{2}(u)~~\text{在}~~\mathbb{R}^{n}\setminus B_{R_0}中, \end{cases} \]的正上解，其中\(q>1,\lambda_{1},\lambda_{2}>0\)，非线性项在原点附近或无穷远处呈现幂次型行为。引入与极值普奇算子相关的有效维数\(\widetilde n_{+}=\frac{\lambda}{\Lambda}(n - 1)+1\)，确定了临界指数\(q_{c}=\frac{\widetilde n_{+}}{\widetilde n_{+}-1}\)，它控制正上解的定性行为。利用此框架，在外部区域建立了尖锐的刘维尔型不存在性定理，并通过梯度指数\(q\)、有效维数\(\widetilde n_{+}\)和非线性耦合之间的相互作用确定了最优不存在区域。在原型情况\(f_{1}(t)=t^{p_{1}}\)，\(f_{2}(t)=t^{p_{2}}\)下，所得条件被证明是最优的。分析是在自然正则性假设\(u,v\in W^{2,p}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus B_{R_0})\)，\(p>n\)下进行的，这是完全非线性一致椭圆方程可用正则性，而非在经典\(C^{2}\)框架内。我们的结果提供了涉及非线性梯度项的半线性椭圆系统的刘维尔理论的完全非线性普奇类似物，并揭示了有效维数在确定临界不存在阈值中的基本作用。
英文摘要
This article investigates positive supersolutions of the fully nonlinear elliptic system \[ \begin{cases} -\mathcal{M}_{λ,Λ}^{+}(D^{2}u)+|\nabla u|^{q} \geqλ_{1}f_{1}(v)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0},\\ -\mathcal{M}_{λ,Λ}^{+}(D^{2}v)+|\nabla v|^{q} \geqλ_{2}f_{2}(u)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0}, \end{cases} \] where $q>1,λ_{1},λ_{2}>0,$ and the nonlinearities exhibit power-type behaviour either near the origin or at infinity. Introducing the effective dimension $\widetilde n_{+}=\fracλΛ(n-1)+1$ associated to extremal Pucci operator, we identify the critical exponent $q_{c}=\frac{\widetilde n_{+}}{\widetilde n_{+}-1},$ which governs the qualitative behaviour of positive supersolutions. Using this framework, we establish sharp Liouville-type nonexistence theorems in exterior domains and determine optimal nonexistence regions through the interaction between the gradient exponent $q,$ the effective dimension $\widetilde n_{+}$ and the nonlinear couplings. In the prototype case $f_{1}(t)=t^{p_{1}},$ $f_{2}(t)=t^{p_{2}}$ the obtained conditions are shown to be optimal. The analysis is carried out under the natural regularity assumption $u,v\in W^{2,p}_{\mathrm{loc}}(\mathbb{R}^{n}\setminus B_{R_0}),$ for $p>n,$ which is the regularity available for fully nonlinear uniformly elliptic equations, rather than within a classical $C^{2}$ framework. Our results provide the fully nonlinear Pucci analogue of the Liouville theory for semilinear elliptic systems involving nonlinear gradient terms and reveal the fundamental role of the effective dimension in determining the critical nonexistence thresholds.
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