Classification of solutions of a critical Hardy–Sobolev operator

Mancini, G.B. John; Fabbri, Isabella; Sandeep, K.

doi:10.1016/j.jde.2005.07.001

Cited by 50 publications

(25 citation statements)

References 15 publications

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“…This critical exponent plays an important role to study the sharp constants and the extremal functions in (2.1) (cf. [24] and [29]). The extremal functions in D 1,2 (R n ) \ {0} of (2.1) can be obtained by investigating the functional…”

Section: Hardy-sobolev Type Equationsmentioning

confidence: 99%

Critical conditions and finite energy solutions of several nonlinear elliptic PDEs inRn

Lei

2015

Journal of Differential Equations

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This paper is concerned with the critical conditions of nonlinear elliptic equations and the corresponding integral equations involving Riesz potentials and Bessel potentials. We show that the equations and some energy functionals are invariant under the scaling transformation if and only if the critical conditions hold. In addition, the Pohozaev identity shows that those critical conditions are the necessary and sufficient conditions for existence of the finite energy positive solutions. Finally, we discuss respectively the existence of the negative solutions of the k-Hessian equations in the subcritical case, critical case and supercritical case. Here the Serrin type critical exponent and the Sobolev type critical exponent play key roles.

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Section: Hardy-sobolev Type Equationsmentioning

confidence: 99%

Critical conditions and finite energy solutions of several nonlinear elliptic PDEs inRn

Lei

2015

Journal of Differential Equations

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“…If α = 2 and p is as above, our Theorem 2 provides a new proof of regularity of extremal function of Hardy-Sobolev inequality, since we prove it through the corresponding integral equation. See [20] for Hölder's regularity of solutions of (4) in case of α = 2. Moreover our Theorem 2 not only proves the regularity of extremal function of (4), but can also be applied to regularity for more general integral equations or corresponding partial differential equations.…”

Section: Is the Greatest Integer Functionmentioning

confidence: 99%

“…Such inequalities of second order have been extensively studied by many authors (see [3,6,7,13,20] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality

Zhu

2011

Calc. Var.

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Let α be a real number satisfying 0 < α < n, 0 ≤ t < α, α * (t) = 2(n−t) n−α . We consider the integral equationwhich is closely related to the Hardy-Sobolev inequality. In this paper, we prove that every positive solution u(x) is radially symmetric and strictly decreasing about the origin by the method of moving plane in integral forms. Moreover, we obtain the regularity of solutions to the following integral equationthat corresponds to a large class of PDEs by regularity lifting method.

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“…The quantitative properties of this type equation are also interesting in critical point theory and nonlinear elliptic equations (cf. [1,2,5,25,27]). …”

Section: Introductionmentioning

confidence: 99%

Asymptotic properties of positive solutions of the Hardy–Sobolev type equations

Lei

2013

Journal of Differential Equations

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Hardy-Sobolev type integral equation Decay rates Integrable solution Bounded decaying solution In this paper, we are concerned with the Lane-Emden type 2m-order PDE with weightwhere n 3, p > 1, m ∈ [1, n/2), σ ∈ (−2m, 0], and the more general Hardy-Sobolev type integral equationthen there is not any positive solution to such an integral equation. Under the assumption of p > n+σ n−α , we obtain that the integrable solution u of the integral equation (i.e. u ∈ L n(p−1)α+σ (R n )) is bounded and decays fast with rate n − α. On the other hand, if the bounded solution is not integrable and decays with some rate, then the rate must be the slow one α+σ p−1 . In addition, the classical solution u of the 2m-order PDE satisfies the integral equation with α = 2m. Therefore, for the 2m-order PDE, all the decay properties above are still true.

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Classification of solutions of a critical Hardy–Sobolev operator

Cited by 50 publications

References 15 publications

Critical conditions and finite energy solutions of several nonlinear elliptic PDEs inRn

Critical conditions and finite energy solutions of several nonlinear elliptic PDEs inRn

Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality

Asymptotic properties of positive solutions of the Hardy–Sobolev type equations

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