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Semi-classical solutions for Kirchhoff type problem with a critical frequency
Part of
- Equations and inequalities involving nonlinear operators
- Elliptic equations and systems
Qilin Xie, Xu Zhang
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 2 / April 2021

Published online by Cambridge University Press:

27 May 2020, pp. 761-798

Print publication:

April 2021
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In the present paper, we consider the following Kirchhoff type problem
$$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$
where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$. As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

The impact of CMEs on the critical frequency of F2-layer ionosphere (foF2)
Alene Seyoum, Nat Gopalswamy, Melessew Nigussie, Nigusse Mezgebe
Journal:

Proceedings of the International Astronomical Union / Volume 15 / Issue S356 / October 2019

Published online by Cambridge University Press:

29 January 2021, pp. 400-402

Print publication:

October 2019
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The ionospheric critical frequency (foF2) from ionosonde measurements at geographic high, middle, and low latitudes are analyzed with the occurrence of coronal mass ejections (CMEs) in long term variability of the solar cycles. We observed trends of monthly maximum foF2 values and monthly averaged values of CME parameters such as speed, angular width, mass, and kinetic energy with respect to time. The impact of CMEs on foF2 is very high at high latitudes and low at low latitudes. The time series for monthly maximum foF2 and monthly-averaged CME speed are moderately correlated at high and middle latitudes.

On the Accuracy of the Timoshenko Beam Theory Above the Critical Frequency: Best Shear Coefficient
J. A. Franco-Villafañe, R. A. Méndez-Sánchez
Journal:

Journal of Mechanics / Volume 32 / Issue 5 / October 2016

Published online by Cambridge University Press:

21 January 2016, pp. 515-518

Print publication:

October 2016
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We obtain values for the shear coefficient both below and above the critical frequency by comparing the results of the Timoshenko beam theory with experimental data published recently. The best results are obtained, by a least-square fitting, when different values of the shear coefficient are used below and above the critical frequency.