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

This article investigates positive supersolutions of the fully nonlinear elliptic system \[ \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}u)+|\nabla u|^{q} \geq\lambda_{1}f_{1}(v)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0},\\ -\mathcal{M}_{\lambda,\Lambda}^{+}(D^{2}v)+|\nabla v|^{q} \geq\lambda_{2}f_{2}(u)~~\text{in}~~\mathbb{R}^{n}\setminus B_{R_0}, \end{cases} \] where $q>1,\lambda_{1},\lambda_{2}>0,$ and the nonlinearities exhibit power-type behaviour either near the origin or at infinity.

Introducing the effective dimension $\widetilde n_{+}=\frac{\lambda}{\Lambda}(n-1)+1$ 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.