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 \(q\in C(\partial\Omega)\), \(h>1\), \(\Delta_\infty^h u=|Du|^{h-3}\Delta_\infty u\).
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 non-decreasing 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 non-increasing 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 boundary-value 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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