Boundary Harnack Inequality and a Priori Estimates of Singular Solutions of Quasilinear Elliptic Equations

Bidaut‐Véron, Marie‐Françoise; Borghol, Rouba; Верон, Лаурент

doi:10.1007/s00526-006-0003-7

Cited by 24 publications

(23 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our results hinge on a uniform Harnack estimates for positive solutions near an isolated Fuchsian type singularity [4,21], and on a ratio limit theorem for the quotients of any such two solutions near a weak Fuchsian singularity [6, Theorem 2.6]. The following statements are formulated for the potentials under consideration.…”

Section: Theorem 24 (Weak Comparison Principle) (Seementioning

confidence: 97%

Isolated singularities of positive solutions of p-Laplacian type equations inRd

Fraas

Pinchover

2013

Journal of Differential Equations

View full text Add to dashboard Cite

Dedicated to the memory of Vitali Liskevich MSC:primary 35B40 secondary 35B09, 35B53, 35J92We study the behavior of positive solutions of p-Laplacian type elliptic equations of the formWe obtain removable singularity theorems for positive solutions near ζ . In particular, using a new three-spheres theorems for certain solutions of the above equation near ζ we prove that if V belongs to a certain Kato class near ζ and p > d (respectively, p < d), then any positive solution u of the equation Q (u) = 0 in a punctured neighborhood of ζ = 0 (respectively, ζ = ∞) is in fact continuous at ζ . Under further assumptions we find the asymptotic behavior of u near ζ .

show abstract

Section: Theorem 24 (Weak Comparison Principle) (Seementioning

confidence: 97%

Isolated singularities of positive solutions of p-Laplacian type equations inRd

Fraas

Pinchover

2013

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…In the proof of (iii) the boundary Harnack inequalities that satisfies any positive solution of (1.4) (see [1]) play a fundamental role. The role of Harnack inequalities has already been a key tool for studying internal isolated singularities for related equations (see [6,14,15,20]).…”

Section: Theorem Let G Be a Continuous Function Such Thatmentioning

confidence: 99%

Boundary singularities of solutions of N-harmonic equations with absorption

Borghol

Верон

2006

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

We study the boundary behaviour of solutions u of − N u + |u| q−1 u = 0 in a bounded smooth domain Ω ⊂ R N subject to the boundary condition u = 0 except at one point, in the range q > N − 1. We prove that if q 2N − 1 such an u is identically zero, while, if N − 1 < q < 2N − 1, u inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.

show abstract

“…The proof below uses a technique involving a scaling argument together with a comparison principle that has been used for example in [5].…”

Section: Minimal Growthmentioning

confidence: 99%

Ground state alternative for p-Laplacian with potential term

2006

View full text Add to dashboard Cite

Let Ω be a domain in R d , d ≥ 2, and 1 < p < ∞. Fix V ∈ L ∞ loc (Ω). Consider the functional Q and its Gâteaux derivative Q ′ given byIn the latter case, v is (up to a multiplicative constant) the unique positive supersolution of the equation Q ′ (u) = 0 in Ω, and one has for Q an inequality of Poincaré type: there exists a positive continuous function W such that for every ψ ∈ C ∞ 0 (Ω) satisfying ψv dx = 0 there exists a constant C > 0 such that C −1 W |u| p dx ≤ Q(u) + C uψ dx p for all u ∈ C ∞ 0 (Ω). As a consequence, we prove positivity properties for the quasilinear operator Q ′ that are known to hold for general subcritical resp. critical second-order linear elliptic operators.

show abstract

Boundary Harnack Inequality and a Priori Estimates of Singular Solutions of Quasilinear Elliptic Equations

Cited by 24 publications

References 16 publications

Isolated singularities of positive solutions of p-Laplacian type equations inRd

Isolated singularities of positive solutions of p-Laplacian type equations inRd

Boundary singularities of solutions of N-harmonic equations with absorption

Ground state alternative for p-Laplacian with potential term

Contact Info

Product

Resources

About