Electron. J. Differential Equations, Vol. 2023 (2023), No. 42, pp. 123.
Viscosity solutions to the infinity Laplacian equation with lower terms
Cuicui Li, Fang Liu
Abstract:
We establish the existence and uniqueness of viscosity solutions to
the Dirichlet problem
$$\displaylines{
\Delta_\infty^h u=f(x,u), \quad \hbox{in } \Omega,\cr
u=q, \quad\hbox{on }\partial\Omega,
}$$
where \(q\in C(\partial\Omega)\), \(h>1\), \(\Delta_\infty^h u=Du^{h3}\Delta_\infty u\).
The operator \(\Delta_\infty u=\langle D^2uDu,Du \rangle\) is the infinity
Laplacian which is strongly degenerate, quasilinear and it is associated with the
absolutely minimizing Lipschitz extension.
When the nonhomogeneous term \(f(x,t)\) is nondecreasing in \(t\), we prove the
existence of the viscosity solution via Perron's method.
We also establish a uniqueness result based on the perturbation analysis of
the viscosity solutions. If the function \(f(x,t)\) is nonpositive (nonnegative)
and nonincreasing in \(t\), we also give the existence of viscosity solutions
by an iteration technique under the condition that the domain has small diameter.
Furthermore, we investigate the existence and uniqueness of viscosity solutions
to the boundaryvalue problem with singularity
$$\displaylines{
\Delta_\infty^h u=b(x)g(u), \quad \hbox{in } \Omega, \cr
u>0, \quad \hbox{in } \Omega, \cr
u=0, \quad \hbox{on }\partial\Omega,
}$$
when the domain satisfies some regular condition.
We analyze asymptotic estimates for the viscosity solution near the boundary.
Submitted January 21, 2023. Published June 25, 2023.
Math Subject Classifications: 35J60, 35J70, 35K55, 35P30.
Key Words: Infinity Laplacian; existence; uniqueness; asymptotic estimate; viscosity solution.
DOI: https://doi.org/10.58997/ejde.2023.42
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Cuicui Li
Department of Mathematics
School of Mathematics and Statistics
Nanjing University of Science and Technology
Nanjing 210094, Jiangsu, China
email: licui1121@njust.edu.cn


Fang Liu
Department of Mathematics
School of Mathematics and Statistics
Nanjing University of Science and Technology
Nanjing 210094, Jiangsu, China
email: sdqdlf78@126.com, liufang78@njust.edu.cn

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