Bessel pairs and optimal Hardy and Hardy–Rellich inequalities

Ghoussoub, Nassif; Moradifam, Amir

doi:10.1007/s00208-010-0510-x

Cited by 141 publications

(170 citation statements)

References 33 publications

(94 reference statements)

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“…On the other hand we shall use certain improved Hardy-Rellich inequalities to prove that the extremal solution is singular in dimensions N 9. Our improve HardyRellich inequalities follow from the recent result of Ghoussoub and Moradifam [12,13] about Hardy and Hardy-Rellich inequalities.…”

Section: Introductionmentioning

confidence: 69%

On the critical dimension of a fourth order elliptic problem with negative exponent

Moradifam

2010

Journal of Differential Equations

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We study the regularity of the extremal solution of the semilinearis a parameter, while τ > 0, β > 0 are fixed constants. It is known that there exists λ * such that for λ > λ * there is no solution while for λ < λ * there is a branch of minimal solutions. Our main result asserts that the extremal solution u * is regular (sup B u * < 1) for N 8 and β, τ > 0 and it is singular (sup B u * = 1) for N 9, β > 0, and τ > 0 with τ β small. Our proof for the singularity of extremal solutions in dimensions N 9 is based on certain improved Hardy-Rellich inequalities.

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Section: Introductionmentioning

confidence: 69%

On the critical dimension of a fourth order elliptic problem with negative exponent

Moradifam

2010

Journal of Differential Equations

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“…Finally, we provide a nonexistence result for inequality (34). We emphasize that inequality (35) has a minimizer and is given by (36).…”

Section: Nonexistence Of Minimizers For Inequality (34)mentioning

confidence: 97%

Some Results Concerning Hardy and Hardy Type Inequalities

Zographopoulos

2014

Handbook of Functional Equations

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We review some recent results concerning functional aspects of the Hardy and Hardy type inequalities. Our main focus is the formulation of such inequalities, for functions having bad behavior at the singularity points. It turns out that Hardy's singularity terms appear in certain cases as a loss to the Hardy's functional, while in other cases are additive to it. Surprisingly, in the latter case, Hardy's functional may be negative. Thus, the validity of the Hardy's inequality is actually based on these singularity terms.We also discuss the two topics: nonexistence of H 1 0 minimizers and improved Hardy-Sobolev inequalities. These topics may be seen as a consequence of the connection of the Hardy and Hardy type inequalities with the Sobolev inequality defined in the whole space.

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“…Moreover for α ∈ [0, n) then γ α = (n − α) 2 /4. These facts are discussed in [9,13,18]. Taking β as in (1.4) makes problem (1.3) invariant with respect to the action of the weighted dilation group…”

Section: Vol 13 (2016) Semilinear Elliptic Equations With Weights 659mentioning

confidence: 99%

Entire Solutions for a Class of Fourth-Order Semilinear Elliptic Equations with Weights

Caldiroli

Cora

2015

Mediterr. J. Math.

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Abstract.We investigate the problem of entire solutions for a class of fourth-order, dilation invariant, semilinear elliptic equations with powertype weights and with subcritical or critical growth in the nonlinear term. These equations define noncompact variational problems and are characterized by the presence of a term containing lower order derivatives, whose strength is ruled by a parameter λ. We can prove existence of entire solutions found as extremal functions for some Rellich-Sobolev type inequalities. Moreover, when the nonlinearity is suitably close to the critical one and the parameter λ is large, symmetry breaking phenomena occur and in some cases the asymptotic behavior of radial and nonradial ground states can be somehow described.Mathematics Subject Classification. 26D10, 47F05.

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Bessel pairs and optimal Hardy and Hardy–Rellich inequalities

Cited by 141 publications

References 33 publications

On the critical dimension of a fourth order elliptic problem with negative exponent

On the critical dimension of a fourth order elliptic problem with negative exponent

Some Results Concerning Hardy and Hardy Type Inequalities

Entire Solutions for a Class of Fourth-Order Semilinear Elliptic Equations with Weights

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