Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Lu, Guozhen; Wang, Peiyong; Zhu, Jiuyi

doi:10.1016/j.anihpc.2012.02.004

Cited by 19 publications

(5 citation statements)

References 25 publications

(29 reference statements)

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“…Remark 2.6. Compared to the singularity and decay estimates of the solutions to elliptic equations that have been obtained in previous studies (e.g., [14]), our result presented here gives a more precise estimate for higher order derivatives of the solution to p-Laplacian equation.…”

Section: Estimates Of Higher Order Derivativessupporting

confidence: 51%

“…The corresponding Liouville type theorems ( e.g. [17,10,8,15,1,14]) are important basis of this method. This method can be briefly described as follows: by proof of contradiction, assuming that an estimate (in terms of the distance to ∂Ω) fails, we could construct an appropriate auxiliary function and use "doubling" property, then the sequence of violating solutions u k will be increasingly large along a sequence of points x k , such that each x k has a suitable neighborhood where the relative growth of u k remains controlled.…”

Section: Proofsmentioning

confidence: 99%

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Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

X¹,

Bao²

2021

Preprint

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In this paper, we apply blow-up analysis to study pointwise a priori estimates for some p-Laplace equations based on Liouville type theorems. With newly developed analysis techniques, we first extend the classical results of interior gradient estimates for the harmonic function to that for the p-harmonic function, i.e., the solution of ∆ p u = 0, x ∈ Ω. We then obtain singularity and decay estimates of the sign-changing solution of Lane-Emden-Fowler type p-Laplace equation −∆ p u = |u| λ−1 u, x ∈ Ω, which are then extended to the equation with general right hand term f (x, u) with certain asymptotic properties. In addition, pointwise estimates for higher order derivatives of the solution to Lane-Emden type p-Laplace equation, in a case of p = 2, are also discussed.

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Section: Estimates Of Higher Order Derivativessupporting

confidence: 51%

Section: Proofsmentioning

confidence: 99%

Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

X¹,

Bao²

2021

Preprint

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“…For the elliptic equations involving either local or nonlocal operators, there have been numerous articles that dedicated to the study of Liouville type theorems, such as [5,15,16,18,19,25,30,33,36] and so on.…”

Section: Introductionmentioning

confidence: 99%

Liouville theorems for fractional parabolic equations

Chen

2021

Preprint

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In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 at infinity with respect to the spacial variables to a polynomial growth on u by constructing auxiliary functions. Then we derive monotonicity for the solutions in a half space R n + × R and obtain some new connections between the nonexistence of solutions in a half space R n + × R and in the whole space R n−1 × R and therefore prove the corresponding Liouville type theorems.To overcome the difficulty caused by the non-locality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of non-local parabolic problems.

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“…We decide not to pursue it in this paper. Finally, we remark that there have been many papers devoted to Liouville theorems for nonnegative solutions of nonlinear polyharmonic equations with the homogeneous Dirichlet boundary condition or homogeneous Navier boundary condition; see Reichel-Weth [40], Lu-Wang-Zhu [38], Chen-Fang-Li [12] and references therein, where they proved that 0 is the unique solution.…”

Section: Introductionmentioning

confidence: 99%

Classification theorems for solutions of higher order boundary conformally invariant problems, I

Sun

Xiong

2016

Journal of Functional Analysis

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In this paper, we prove that nonnegative polyharmonic functions on the upper half space satisfying a conformally invariant nonlinear boundary condition have to be the "polynomials plus bubbles" form. The nonlinear problem is motivated by the recent studies of boundary GJMS operators and the Q-curvature in conformal geometry. The result implies that in the conformal class of the unit Euclidean ball there exist metrics with a single singular boundary point which have flat Q-curvature and constant boundary Q-curvature. Moreover, all of such metrics are classified. This phenomenon differs from that of boundary singular metrics which have flat scalar curvature and constant mean curvature, where the singular set contains at least two points. A crucial ingredient of the proof is developing an approach to separate the higher order linear effect and the boundary nonlinear effect so that the kernels of the nonlinear problem are captured.

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Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Cited by 19 publications

References 25 publications

Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

Pointwise A Priori Estimates for Solutions to Some p-Laplacian Equations

Liouville theorems for fractional parabolic equations

Classification theorems for solutions of higher order boundary conformally invariant problems, I

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