Liouville theorems for infinity Laplacian with gradient and KPP type\n equation

2020/03/22 by Anup Biswas, Biswas, Anup, Hoang‐Hung Vo +1 · 4 citations
Computer Science · Mathematics · #35B65 #35J60 #35J70 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2003.10056

openalex publication_date 2020/03/22 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove new Liouville type results for a nonlinear equation\ninvolving infinity Laplacian with gradient of the form
Delta^
gamma_
infty u\n+ q(x)
cdot
nablau |
nablau|^2-
gamma + f(x, u)
,=
,0
quad
textin
;\n
mathbbR^d, where \γ\∈ [0, 2] and \Δ^\γ_\∞ is a\n(3-\γ)-homogeneous operator associated with the infinity Laplacian. Under\nthe assumptions liminf_|x|\→\∞\lim_s\→0f(x,s)/s^3-\γ>0 and\nq is a continuous function vanishing at infinity, we construct a positive\nbounded solution to the equation and if f(x,s)/s^3-\γ decreasing in\ns, we further obtain the uniqueness by improving sliding method for infinity\nLaplacian operator with nonlinear gradient. Otherwise, if\n limsup_|x|\→\∞\sup_[\δ1,\δ2]f(x,s)<0, then nonexistence\nresult holds provided additionally some suitable conditions. To this aim, we\ndevelop novel techniques to overcome the difficulties stemming from the\ndegeneracy of infinity Laplacian and nonlinearity of the gradient term. Our\napproach is based on a new regularity result, the strong maximum principle, and\nHopf's lemma for infinity Laplacian involving gradient and potential. We also\nconstruct some examples to illustrate our results. We further investigate some\ndeeper qualitative properties of the principal eigenvalue of the corresponding\nnonlinear operator
Delta^
gamma_
infty u + q(x)
cdot
nablau\n|
nablau|^2-
gamma + c(x)u^3-
gamma, with Dirichlet boundary condition\nin smooth bounded domains, which may be of independent interest. The results\nobtained here could be considered as sharp extension of the Liouville type\nresults obtained in [1, 2, 11, 24, 48, 52].\n

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• Regularity estimates for fully nonlinear dead-core problems with a Hamiltonian Term
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