On the Liouville property for fully nonlinear equations with superlinear first-order terms

We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is F ( x , D 2 u stretchy="false">) ≥ H i F(x,D^2u)\geq H_i(x,u,Du) alttext="double-struck R Superscript N"> R N encoding="application/x-tex">\mathbb {R}^N , where i"> encoding="application/x-tex">H_i has superlinear growth the gradient variable. After a brief survey on existing literature, we discuss validity or failure property model cases 1 equals q plus StartAbsoluteValue EndAbsoluteValue gamma"> 1 = q + stretchy="false">| γ encoding="application/x-tex">H_1(u,Du)=u^q+|Du|^\gamma 2 encoding="application/x-tex">H_2(u,Du)=u^q|Du|^\gamma and 3 plus-or-minus gamma minus b dot u"> 3 ±<!-- ± <mml:mo>−<!-- − <mml:mi>b ⋅<!-- ⋅ encoding="application/x-tex">H_3(x,u,Du)=\pm u^q|Du|^\gamma -b(x)\cdot Du alttext="q 0"> 0 encoding="application/x-tex">q\geq 0 alttext="gamma greater-than 1"> &gt; encoding="application/x-tex">\gamma &gt;1 alttext="b"> encoding="application/x-tex">b suitable velocity field. Several counterexamples open problems are thoroughly discussed.