Liouville property of infinity Laplacian related to the eigenvalue problems with gradient

Anup Biswas, Hoang-Hung Vo

Published 2020-03-23Version 1

We study the Liouville type results for a nonlinear equation involving infinity Laplacian with gradient of the form $$\Delta^\gamma_\infty u + q(x)\cdot \nabla{u} |\nabla{u}|^{2-\gamma} + f(x, u)\,=\,0\quad \text{in}\; \mathbb{R}^d,$$ 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 can 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 give a major contribution to the literature besides the Liouville type results obtained by Savin [45] and Ara\'{u}jo et.\ al.\ [1] and a counterpart of the uniqueness obtained by Lu and Wang [36,37] for sign-changing $f$. We believe that the method used in this paper can be applied for the wide range of equations involving infinity Laplacian and gradient term.