Lie Symmetries and Criticality of Semilinear Differential Systems

Bozhkov, Yuri; Mitidieri, Enzo

doi:10.3842/sigma.2007.053

Cited by 10 publications

(11 citation statements)

References 31 publications

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“…The critical exponents can be also viewed as numbers which divide the existence and nonexistence cases for various differential equations and systems. The notion of criticality of differential equations, its relations to scaling transformations and to the present approach is discussed in another work [2].…”

Section: )mentioning

confidence: 99%

The Noether Approach to Pokhozhaev’s Identities

Bozhkov

Mitidieri

2007

MedJM

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Section: )mentioning

confidence: 99%

The Noether Approach to Pokhozhaev’s Identities

Bozhkov

Mitidieri

2007

MedJM

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“…In [Bozhkov(2005)] this fact was firstly discussed. Later, in [Bozhkov and Mitidieri(2007)], this point was retaken and many examples were analysed. A considerable number of the examples discussed were related with partial differential equations.…”

Section: Main Results and Preliminary Discussionmentioning

confidence: 99%

Symmetry analysis of a class of autonomous even-order ordinary differential equations

Silva

Freire

2015

IMA Journal of Applied Mathematics

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A class of autonomous, even-order ordinary differential equations is discussed from the point of view of Lie symmetries. It is shown that for a certain power nonlinearity, the Noether symmetry group coincides with the Lie point symmetry group. First integrals are established and exact solutions are found. Furthermore, this paper complements, for the one-dimensional case, some results in the literature of Lie group analysis of poliharmonic equations and Noether symmetries obtained in the last twenty years. In particular, it is shown that the exceptional negative power discovered in [Bokhari, A. H., Mahomed, F. M. and Zaman, F. D. (2010) Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation. J. Math. Phys., 51, 053517] is a member of a one-parameter family of exceptional powers in which the Lie symmetry group coincides with the Noether symmetry group. Lie point symmetries, Noether symmetries, first integrals, exact solutions, ordinary differential equations, power nonlinearities.

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“…In this procedure one chooses critical values of the involved parameters. (See [5] on the last point.) In a subsequent work [6] we have applied this method to some semilinear partial differential equations and systems on Riemannian manifolds.…”

Section: Introductionmentioning

confidence: 99%

Conformal Killing Vector Fields and Rellich Type Identities on Riemannian Manifolds, II

Bozhkov

Mitidieri

2011

Mediterr. J. Math.

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We propose a general Noetherian approach to Rellich integral identities. Using this method we obtain a higher order Rellich type identity involving the polyharmonic operator on Riemannian manifolds admitting homothetic transformations. Then we prove a biharmonic Rellich identity in a more general context. We also establish a nonexistence result for semilinear systems involving biharmonic operators.Mathematics Subject Classification (2010). 35J50, 35J20, 35J60.

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Lie Symmetries and Criticality of Semilinear Differential Systems

Cited by 10 publications

References 31 publications

The Noether Approach to Pokhozhaev’s Identities

The Noether Approach to Pokhozhaev’s Identities

Symmetry analysis of a class of autonomous even-order ordinary differential equations

Conformal Killing Vector Fields and Rellich Type Identities on Riemannian Manifolds, II

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