A strong maximum principle and a compact support principle for singular elliptic inequalities

Pucci, Patrizia; Serrin, James; Zou, Henghui

doi:10.1016/s0021-7824(99)00030-6

Cited by 97 publications

(117 citation statements)

References 7 publications

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“…The operators L ϕ,h may be viewed as the natural, intrinsic generalization to Riemannian manifolds of the fully quasilinear singular elliptic operators considered by Pucci Serrin and Zou (see [21], [19], [20]). …”

Section: Where : T * M → T M Denotes the Musical Isomorphism So Thatmentioning

confidence: 99%

“…We note that we will apply our comparison principle to functions that are not necessarily C 1 (Ω), most notably to radial functions on the underlying manifold, which are in general only Lipschitz. An alternative proof of the comparison principe valid for Lipschitz functions may be obtained by using Lemma 1.1 to adapt to the case of the operator L ϕ,h the comparison principle for the ϕ-Laplacian contained in [18, Proposition 6.1] (see also [21,Lemma 3]). …”

Section: Lemma 12 Assume That ϕ and H Satisfy Conditions (01) (I)-mentioning

confidence: 99%

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Some non-linear function theoretic properties of Riemannian manifolds

Pigola¹,

Rigoli²,

Setti³

2006

Rev. Mat. Iberoam.

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We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the p-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators. IntroductionThe starting point of the present note is the circle of ideas in classical potential theory, which relate the parabolicity and stochastic completeness of a manifold on one hand, and their function theoretic counterparts on the other, and in particular results due to Khas'minskii which provide sufficient conditions for a manifold to be parabolic, respectively stochastically complete. These ideas allow also to establish comparison theorems with model manifolds in the sense of Greene and Wu under curvature conditions. It was recently shown by the authors that the parabolicity/stochastic completeness of a manifold may be described in terms of suitable versions of a global weak maximum principle. Explicitly, M is parabolic (resp. stochastically complete) if and only if for every u ∈ C 2 (M ), u bounded above, and for every η > 0 we have inf {u>sup u−η} ∆u < 0 (resp. ≤ 0) (see [17] and [18]).

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Section: Where : T * M → T M Denotes the Musical Isomorphism So Thatmentioning

confidence: 99%

Section: Lemma 12 Assume That ϕ and H Satisfy Conditions (01) (I)-mentioning

confidence: 99%

Some non-linear function theoretic properties of Riemannian manifolds

Pigola¹,

Rigoli²,

Setti³

2006

Rev. Mat. Iberoam.

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“…We herein introduce the weak version of the maximum principle proved by Pucci et al [17]. LEMMA 2.1.…”

Section: Preliminariesmentioning

confidence: 99%

Existence Results to Some Quasilinear Elliptic Problems With Hardy Potential and Absorption Term

Wang¹,

Zhang²

2014

Kyushu J. Math.

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Abstract. This article is devoted to dealing with existence of a solution to a quasilinear elliptic problem with the Hardy potential term u p−1 /|x| p and critical growth in the gradient |∇u| p as an absorption term. By considering the interaction between the two terms in the equation, we prove that there exists a positive solution to the problem. Moreover, the existence result for a wider class of weight functions can also be obtained.

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“…[10,18,17,19,21,22,23,24,29] and the references therein. Maximum principles often essentially express that the studied equation is a qualitatively reliable model of the underlying real phenomenon, e.g.…”

Section: Introductionmentioning

confidence: 99%

A maximum principle for some nonlinear cooperative elliptic PDE systems with mixed boundary conditions

Karátson

2016

Journal of Mathematical Analysis and Applications

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One of the classical maximum principles state that any nonnegative solution of a proper elliptic PDE attains its maximum on the boundary of a bounded domain. We suitably extend this principle to nonlinear cooperative elliptic systems with diagonally dominant coupling and with mixed boundary conditions. One of the consequences is a preservation of nonpositivity, i.e. if the coordinate functions or their fluxes are nonpositive on the Dirichlet or Neumann boundaries, respectively, then they are all nonpositive on the whole domain as well. Such a result essentially expresses that the studied PDE system is a qualitatively reliable model of the underlying real phenomena, such as proper reaction-diffusion systems in chemistry.

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A strong maximum principle and a compact support principle for singular elliptic inequalities

Cited by 97 publications

References 7 publications

Some non-linear function theoretic properties of Riemannian manifolds

Some non-linear function theoretic properties of Riemannian manifolds

Existence Results to Some Quasilinear Elliptic Problems With Hardy Potential and Absorption Term

A maximum principle for some nonlinear cooperative elliptic PDE systems with mixed boundary conditions

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