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1

Entire solutions for some critical equations in the Heisenberg group

Pucci Patrizia, Temperini Letizia

Opuscula Mathematica

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2022

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Vol. 42, no. 2

279--303

EN

We complete the study started in the paper [P. Pucci, L. Temperini, On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), Paper no. 007], giving some applications of its abstract results to get existence of solutions of certain critical equations in the entire Heinseberg group. In particular, different conditions for existence are given for critical horizontal p-Laplacian equations.

2

Existence and multiplicity results for quasilinear equations in the Heisenberg group

Pucci Patrizia

Opuscula Mathematica

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2019

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Vol. 39, no. 2

247--257

EN

In this paper we complete the study started in [Existence of entire solutions for quasilinear equations in the Heisenberg group, Minimax Theory Appl. 4 (2019)] on entire solutions for a quasilinear equation [formula] in Hn, depending on a real parameter λ, which involves a general elliptic operator A in divergence form and two main nonlinearities. Here, in the so called sublinear case, we prove existence for all λ > 0 and, for special elliptic operators A, existence of infinitely many solutions [formula].

3

Existence of solution of sub-elliptic equations on the Heisenberg group with critical growth and double singularities

Chen J., Rocha E. M.

Opuscula Mathematica

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2013

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Vol. 33, no. 2

237--254

EN

For a class of sub-elliptic equations on Heisenberg group HN with Hardy type singularity and critical nonlinear growth, we prove the existence of least energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math.118 (2005), 35–252].

4

Do non-strictly stable laws on positively graduated simply connected nilpotent lie groups lie in their own domain of normal attraction?

Neuenschwander D.

Probability and Mathematical Statistics

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2012

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Vol. 32, Fasc. 2

189--202

EN

In the classical case of the real line, it is clear from the very definition that non-degenerate stable laws always belong to their own domain of normal attraction. The question if the analogue of this is also true for positively graduated simply connected nilpotent Lie groups (a natural framework for the generalization of the concept of stability to the non-commutative case) turns out to be non-trivial. The reason is that, in this case, non-strict stability is defined in terms of generating distributions of continuous one-parameter convolution semigroups rather than just for the laws themselves. We show that the answer is affirmative for non-degenerate (not necessarily strictly) α-dilation-stable laws on simply connected step 2-nilpotent Lie groups (so, e.g., all Heisenberg groups and all so-called groups of type H; cf. Kaplan [6]) if α∈]0; 1[ ∪ ]1; 2]. The proof generalizes to positively graduated simply connected Lie groups which are nilpotent of higher step if α∈[0; 1].

5

Bertrand mate of timelike biharmonic legendre curves in Lorentzian Heisenberg group Heis³

Turhan E., Körpinar T.

Demonstratio Mathematica

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2011

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Vol. 44, nr 2

273-283

EN

In this paper, we study Bertrand mate of timelike biharmonic Legendre curves in the Lorentzian Heisenberg group Heis³. We characterize timelike biharmonic Legendre curves in terms of their curvature and torsion. Moreover, we obtain the position vectors of timelike biharmonic Legendre curves and we construct parametric equations of Bertrand mate of timelike biharmonic Legendre curves in the Lorentzian Heisenberg group Heis³.

6

Characterize on the Heisenberg group with left invariant Lorentzian metric

Turhan T., Korpinar T.

Demonstratio Mathematica

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2009

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Vol. 42, nr 2

423-428

EN

In this paper, we consider the biharmonicity conditions for maps between Riemannian manifolds and we characterize non-geodesic biharmonic curve in Heisenberg group H3 which is endowed with left invariant Lorentzian metric.