A note on the Liouville method applied to elliptic eventually degenerate fully nonlinear equations governed by the Pucci operators and the Keller–Osserman condition

Abstract

This paper is devoted to the study of the viscosity solutions of $$\begin{array}{l}\mathbb{F}({\rm D}^{2}u,u,x)+f=0\end{array}$$ in the whole space \({\mathbb{R}^{\rm N}}\) , under suitable structural assumptions on \({\mathbb{F}}\) involving the Pucci extremal operators for the leading part and the Keller–Osserman condition on the zeroth order term. By means a kind of Liouville method we prove that uniqueness of solutions holds. Therefore a unique growth at infinity is compatible with the structure of the equations. Furthermore the knowledge of these growth at infinity of solutions is not required a priori. Due to the presence of a strong absorption term, the Liouville Theorem proved is independent on the dimension.