Harmonic Radius and Concentration of Energy; Hyperbolic Radius and Liouville’s Equations $\Delta U = e^U $ and $\Delta U = U^{\tfrac{{n + 2}}{{n - 2}}} $

Bandle, Catherine; Flucher, Martin

doi:10.1137/1038039

Cited by 106 publications

(140 citation statements)

References 45 publications

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“…The boundary of the core A := {v > 1} is the free boundary for (1.9). Problem (1.9) arises in plasma physics [3,12,20]. The corresponding problem in two dimensions has been studied in [3]; the case when p = 1 by Caffarelli and Friedman [12].…”

Section: J Weimentioning

confidence: 99%

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Multiple Condensations for a Nonlinear Elliptic Equation With Sub-Critical Growth and Critical Behaviour

Wei

2001

Proceedings of the Edinburgh Mathematical Society

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We consider the following nonlinear elliptic equationswhere u + = max(u, 0), M is a prescribed constant, and Ω is a bounded and smooth domain in, Ω = B R (0), the above problem has a continuum of solutions. The case when M > M (N ) * is referred to as supercritical in the literature. We show that for M near KM (N ) * , K > 1, there exist solutions with multiple condensations in Ω. These concentration points are non-degenerate critical points of a function related to the Green's function.

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Section: J Weimentioning

confidence: 99%

“…Even though it has subcritical growth, it has critical behaviour. Wolansky [39] [3]. ) Wolansky [39] proved the following theorem.…”

Section: Introductionmentioning

confidence: 95%

Multiple Condensations for a Nonlinear Elliptic Equation With Sub-Critical Growth and Critical Behaviour

Wei

2001

Proceedings of the Edinburgh Mathematical Society

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“…The harmonic radius r(y) of D at y is defined by [6] [14] proved that the lowest eigenvalue λ 1 of the Dirichlet Laplacian satisfies λ 1 (D) ≤ λ 1 (B max r ).…”

Section: The Regular Part H(x Y) Is Harmonic In Both Variables Its mentioning

confidence: 99%

“…If D is simply connected, then the function U (y) := − log(r 0 (y)) is a solution of Liouville's equation [6]:…”

Section: The Spherical (Hyperbolic) Radius R ± and The Euclidean Harmmentioning

confidence: 99%

Green’s function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature

Bandle¹,

Brillard

Flucher³

1998

Trans. Amer. Math. Soc.

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Abstract. We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green's function of the corresponding LaplaceBeltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.

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“…Note that we can integrate the gradient estimate to obtain an improved estimate for ψ(x, x). This is implicit in the work [Bandle and Flucher 1996].…”

Section: A Technical Lemmamentioning

confidence: 96%

Deformation retracts to the fat diagonal and applications to the existence of peak solutions of nonlinear elliptic equations

Dancer¹,

Hillman²,

Pistoia³

2012

Pacific J. Math.

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Harmonic Radius and Concentration of Energy; Hyperbolic Radius and Liouville’s Equations $\Delta U = e^U $ and $\Delta U = U^{\tfrac{{n + 2}}{{n - 2}}} $

Cited by 106 publications

References 45 publications

Multiple Condensations for a Nonlinear Elliptic Equation With Sub-Critical Growth and Critical Behaviour

Multiple Condensations for a Nonlinear Elliptic Equation With Sub-Critical Growth and Critical Behaviour

Green’s function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature

Deformation retracts to the fat diagonal and applications to the existence of peak solutions of nonlinear elliptic equations

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