Liouville Theorems for infinity Laplacian with gradient and KPP type equation

Abstract

In this paper, we prove several Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$\Delta^\gamma_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-\gamma} + f(x, u)\,=\,0\quad \text{in}\; \Rd,$$ where $\gamma\in [0, 2]$ and $\Delta^\gamma_\infty$ is a $(3-\gamma)$-homogeneous operator associated with the infinity Laplacian. Under the 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 positive bounded solution to the equation and if $f(x,s)/s^{3-\gamma}$ decreasing in $s$, we also obtain the uniqueness. While, if $\limsup_{|x|\to\infty}\sup_{[\delta_1,\delta_2]}f(x,s)<0$, then nonexistence result holds provided additionally some suitable conditions. To this aim, we develop new technique to overcome the degeneracy of infinity Laplacian and nonlinearity of gradient term. Our approach is based on a new regularity result, the strong maximum principle, and Hopf's lemma for infinity Laplacian involving gradient and potential. We also construct some examples to illustrate our results. We further study the related Dirichlet principal eigenvalue of the corresponding nonlinear operator $$\Delta^\gamma_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-\gamma} + c(x)u^{3-\gamma},$$ in smooth bounded domains, which may be considered as of independent interest. Our results could be seen as the extension of Liouville type results obtained by Savin \cite{S1} and Ara\'{u}jo et.\ al.\ \cite{ALT} and a counterpart of the uniqueness obtained by Lu and Wang \cite{LW2008,LW2008a} for sign-changing $f$.