genus theory
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2020 ◽  
Vol 72 (6) ◽  
pp. 978-989
Author(s):  
S. Taarabti ◽  
Z. El Allali ◽  
K. Ben Haddouch


Author(s):  
Peter Koymans ◽  
Carlo Pagano

Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.



2020 ◽  
Vol 72 (6) ◽  
pp. 842-851
Author(s):  
S. Taarabti ◽  
Z. El Allali ◽  
K. Ben Haddouch

UDC 517.9 The paper deals with the existence and multiplicity of nontrivial weak solutions for the 𝓅 ( x ) -Kirchhoff-type problem, u = Δ u = 0 o n ∂ Ω . By using variational approach and Krasnoselskii’s genus theory, we prove the existence and multiplicity of solutions for the 𝓅 ( x ) -Kirchhoff-type equation.



2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Jie Yang ◽  
Haibo Chen ◽  
Senli Liu

We consider the following double phase problem with variable exponents: −div∇upx−2∇u+ax∇uqx−2∇u=λfx,u in Ω,u=0, on ∂Ω. By using the mountain pass theorem, we get the existence results of weak solutions for the aforementioned problem under some assumptions. Moreover, infinitely many pairs of solutions are provided by applying the Fountain Theorem, Dual Fountain Theorem, and Krasnoselskii’s genus theory.



2020 ◽  
Vol 207 ◽  
pp. 423-459
Author(s):  
Georges Gras
Keyword(s):  




2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Xianzhong Yao ◽  
Xia Li ◽  
Fuchen Zhang ◽  
Chunlai Mu

In this paper, we study the following Schrödinger-Poisson equations −Δu+u+ϕu=u5+λaxup−1u,x∈ℝ3,−Δϕ=u2,x∈ℝ3, where the parameter λ>0 and p∈0,1. When the parameter λ is small and the weight function ax fulfills some appropriate conditions, we admit the Schrödinger-Poisson equations possess infinitely many negative energy solutions by using a truncation technology and applying the usual Krasnoselskii genus theory. In addition, a byproduct is that the set of solutions is compact.



2019 ◽  
Vol 15 (07) ◽  
pp. 1437-1447 ◽  
Author(s):  
Siham Aouissi ◽  
Moulay Chrif Ismaili ◽  
Mohamed Talbi ◽  
Abdelmalek Azizi

Let [Formula: see text] with [Formula: see text] a cube-free positive integer. Let [Formula: see text] be the 3-class group of k. With the aid of genus theory, arithmetic properties of the pure cubic field [Formula: see text] and some results on the 3-class group [Formula: see text], we determine all integers [Formula: see text] such that [Formula: see text].



2018 ◽  
Vol 37 (4) ◽  
pp. 187-203
Author(s):  
Mohammed Massar ◽  
Ahmed Hamydy ◽  
Najib Tsouli

This paper is devoted to the existence of solutions for a class of Kirchhoff type systems involving critical exponents. The proof of the main results is based on  concentration compactness principle related to critical elliptic systems due to Kang combined with genus theory.



2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Jing Li ◽  
Caisheng Chen

We focus on the following elliptic system with critical Sobolev exponents:  -div⁡∇up-2∇u+m(x)up-2u=λup⁎-2u+(1/η)Gu(u,v),  x∈RN; -div⁡∇vq-2∇v+n(x)vq-2v=μvq⁎-2v+(1/η)Gv(u,v),  x∈RN; u(x)>0,v(x)>0,  x∈RN, where μ,λ>0,1<p≤q<N, either η∈(1,p) or η∈(q,p⁎), and critical Sobolev exponents p⁎=pN/(N-p) and q⁎=qN/(N-q). Conditions on potential functions m(x),n(x) lead to no compact embedding. Relying on concentration-compactness principle, mountain pass lemma, and genus theory, the existence of solutions to the elliptic system with η∈(q,p⁎) or η∈(1,p) will be established.



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