The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian

Liu, Fang

doi:10.3934/dcdsb.2021155

Cited by 4 publications

(2 citation statements)

References 33 publications

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“…One can see [36] for more uniqueness results of infinity Laplacian equations. We direct the reader to [27,28,29,30,31,33,36,37,39,42,44] and the references therein for the ∞-Laplacian operator.…”

Section: Introductionmentioning

confidence: 99%

Viscosity solutions to the infinity Laplacian equation with lower terms

Li¹,

Liu²

2023

ejde

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We establish the existence and uniqueness of viscosity solutions tothe 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.

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Section: Introductionmentioning

confidence: 99%

Viscosity solutions to the infinity Laplacian equation with lower terms

Li¹,

Liu²

2023

ejde

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“…when the right hand side ( ) , f x t is non-decreasing in t and has one sign. In addition, it is also necessary to prove the comparison principle during the studies of the Dirichlet eigenvalue problem related to the infinity Laplacian, see for example [28] [29] [30].…”

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Uniqueness of Viscosity Solutions to the Dirichlet Problem Involving Infinity Laplacian

Sun,

Liu

2023

APM

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In this paper, we study the Dirichlet boundary value problem involving the highly degenerate and h-homogeneous quasilinear operator associated with the infinity Laplacian, where the right hand side term is ( ) ( ) , , n F x t p C ∈ Ω × ×   and the boundary value is ( ) C ϕ ∈ ∂Ω . First, we establish the comparison principle by the double variables method based on the viscosity solutions theory for the general equation ( ) , , h u F x u Du ∞ ∆ = in Ω . We propose two different conditions for the right hand side ( ) , , F x u Du and get the comparison principle results under different conditions by making different perturbations. Then, we obtain the uniqueness of the viscosity solution to the Dirichlet boundary value problem by the comparison principle. Moreover, we establish the local Lipschitz continuity of the viscosity solution.

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Viscosity Solutions to the Infinity Laplacian Equation With Singular Nonlinear Terms

LIU,

SUN

2024

J. Aust. Math. Soc.

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In this paper, we study the singular boundary value problem $$ \begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$ where $\lambda>0$ is a parameter, $h>1$ and $\Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator related to the infinity Laplacian. The nonlinear term $f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is a continuous function and may exhibit singularity at $t\rightarrow 0^{+}$ . We establish the comparison principle by the double variables method for the general equation $\Delta _\infty ^h u=F(x,u,Du)$ under some conditions on the term $F(x,t,p)$ . Then, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron’s method and the comparison principle. Finally, we obtain the existence result in the entire Euclidean space by the approximation procedure. In this procedure, we also establish the local Lipschitz continuity of the viscosity solution.

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The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian

Abstract: In this paper, we investigate a weighted Dirichlet eigenvalue problem for a class of degenerate operators related to the h degree homogeneous p-LaplacianHere a(x) is a positive continuous bounded function in the closure of Ω ⊂ R n (n ≥ 2), h > 1, 2 < p < ∞, and ∆ N p u =

Cited by 4 publications

References 33 publications

Viscosity solutions to the infinity Laplacian equation with lower terms

Viscosity solutions to the infinity Laplacian equation with lower terms

Uniqueness of Viscosity Solutions to the Dirichlet Problem Involving Infinity Laplacian

Viscosity Solutions to the Infinity Laplacian Equation With Singular Nonlinear Terms

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