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In this paper, we prove new Liouville type results for a nonlinear equationinvolving infinity Laplacian with gradient of the form $$\Delta^\gamma_\infty u+ q(x)\cdot \nabla{u} |\nabla{u}|^{2-\gamma} + f(x, u)\,=\,0\quad \text{in}\;\mathbb{R}^d,$$ where $\gamma\in [0, 2]$ and $\Delta^\gamma_\infty$ is a$(3-\gamma)$-homogeneous operator associated with the infinity Laplacian. Underthe assumptions $\liminf_{|x|\to\infty}\lim_{s\to0}f(x,s)/s^{3-\gamma}>0$ and$q$ is a continuous function vanishing at infinity, we construct a positivebounded solution to the equation and if $f(x,s)/s^{3-\gamma}$ decreasing in$s$, we further obtain the uniqueness by improving sliding method for infinityLaplacian operator with nonlinear gradient. Otherwise, if$\limsup_{|x|\to\infty}\sup_{[\delta_1,\delta_2]}f(x,s)<0$, then nonexistenceresult holds provided additionally some suitable conditions. To this aim, wedevelop novel techniques to overcome the difficulties stemming from thedegeneracy of infinity Laplacian and nonlinearity of the gradient term. Ourapproach is based on a new regularity result, the strong maximum principle, andHopf's lemma for infinity Laplacian involving gradient and potential. We alsoconstruct some examples to illustrate our results. We further investigate somedeeper qualitative properties of the principal eigenvalue of the correspondingnonlinear operator $$\Delta^\gamma_\infty u + q(x)\cdot \nabla{u}|\nabla{u}|^{2-\gamma} + c(x)u^{3-\gamma},$$ with Dirichlet boundary conditionin smooth bounded domains, which may be of independent interest. The resultsobtained here could be considered as sharp extension of the Liouville typeresults obtained in [1, 2, 11, 24, 48, 52].

Liouville theorems for infinity Laplacian with gradient and KPP type equation