Some remarks on gradient-type systems on the Heisenberg group

D'Onofrio, Luigi; Bisci, Giovanni Molica

doi:10.1080/17476933.2019.1565408

Cited by 3 publications

(1 citation statement)

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“…The existence of weak solutions for subelliptic problems set on ψ has been investigated in [9,16,20] by employing symmetries. The main proofs, crucially based on the Palais Principle, are obtained by developing a suitable algebraic procedure on the unitary group U(n) := U (n) × {id} that fits well with the approach developed along the present paper.…”

Section: Lack Of Compactness and Symmetries On Unbounded Domainsmentioning

confidence: 99%

A Flower-Shape Geometry and Nonlinear Problems on Strip-Like Domains

2020

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In the present paper, we show how to define suitable subgroups of the orthogonal group $${O}(d-m)$$ O ( d - m ) related to the unbounded part of a strip-like domain $$\omega \times {\mathbb {R}}^{d-m}$$ ω × R d - m with $$d\ge m+2$$ d ≥ m + 2 , in order to get “mutually disjoint” nontrivial subspaces of partially symmetric functions of $$H^1_0(\omega \times {\mathbb {R}}^{d-m})$$ H 0 1 ( ω × R d - m ) which are compactly embedded in the associated Lebesgue spaces. As an application of the introduced geometrical structure, we prove (existence and) multiplicity results for semilinear elliptic problems set in a strip-like domain, in the presence of a nonlinearity which either satisfies the classical Ambrosetti–Rabinowitz condition or has a sublinear growth at infinity. The main theorems of this paper may be seen as an extension of existence and multiplicity results, already appeared in the literature, for nonlinear problems set in the entire space $${\mathbb {R}}^d$$ R d , as for instance, the ones due to Bartsch and Willem. The techniques used here are new.

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Section: Lack Of Compactness and Symmetries On Unbounded Domainsmentioning

confidence: 99%

A Flower-Shape Geometry and Nonlinear Problems on Strip-Like Domains

2020

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Multiplicity of solutions for the noncooperative Choquard-Kirchhoff system involving Hardy-Littlewood-Sobolev critical exponent on the Heisenberg group

Sun

Yang

Song

2022

Rend. Circ. Mat. Palermo, II. Ser

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On Nonlinear Biharmonic Problems on the Heisenberg Group

Zuo

Taarabti

et al. 2022

Symmetry

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We investigate the boundary value problem for biharmonic operators on the Heisenberg group. The inherent features of Hn make it an appropriate environment for studying symmetry rules and the interaction of analysis and geometry with manifolds. The goal of this paper is to prove that a weak solution for a biharmonic operator on the Heisenberg group exists. Our key tools are a version of the Mountain Pass Theorem and the classical variational theory. This paper will be of interest to researchers who are working on biharmonic operators on Hn.

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Some remarks on gradient-type systems on the Heisenberg group

Cited by 3 publications

References 34 publications

A Flower-Shape Geometry and Nonlinear Problems on Strip-Like Domains

A Flower-Shape Geometry and Nonlinear Problems on Strip-Like Domains

Multiplicity of solutions for the noncooperative Choquard-Kirchhoff system involving Hardy-Littlewood-Sobolev critical exponent on the Heisenberg group

On Nonlinear Biharmonic Problems on the Heisenberg Group

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