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26078 results in Abstract analysis

Page 130 of 1304

  • On the fractional Lazer-McKenna conjecture with critical growth

  • Part of
  • Qi Li, Shuangjie Peng
  • Journal:
    Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 4 / August 2022
    Published online by Cambridge University Press:
    11 August 2021, pp. 879-911
    Print publication:
    August 2022

  • This paper deals with the following fractional elliptic equation with critical exponent

    \[ \begin{cases} \displaystyle (-\Delta )^{s}u=u_{+}^{2_{s}^{*}-1}+\lambda u-\bar{\nu}\varphi_{1}, & \text{in}\ \Omega,\\ \displaystyle u=0, & \text{in}\ {{\mathfrak R}}^{N}\backslash \Omega, \end{cases}\]
    where $\lambda$, $\bar {\nu }\in {{\mathfrak R}}$, $s\in (0,1)$, $2^{*}_{s}=({2N}/{N-2s})\,(N>2s)$, $(-\Delta )^{s}$ is the fractional Laplace operator, $\Omega \subset {{\mathfrak R}}^{N}$ is a bounded domain with smooth boundary and $\varphi _{1}$ is the first positive eigenfunction of the fractional Laplace under the condition $u=0$ in ${{\mathfrak R}}^{N}\setminus \Omega$. Under suitable conditions on $\lambda$ and $\bar {\nu }$ and using a Lyapunov-Schmidt reduction method, we prove the fractional version of the Lazer-McKenna conjecture which says that the equation above has infinitely many solutions as $|\bar \nu | \to \infty$ .

  • On a reaction–diffusion system modelling infectious diseases without lifetime immunity

  • Part of
  • HONG-MING YIN
  • Journal:
    European Journal of Applied Mathematics / Volume 33 / Issue 5 / October 2022
    Published online by Cambridge University Press:
    09 August 2021, pp. 803-827

  • In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

A Course of Modern Analysis
  • 5th edition
  • E. T. Whittaker, G. N. Watson
  • Edited by Victor H. Moll
  • Published online:
    07 August 2021
    Print publication:
    26 August 2021
  • This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902. Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge. The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis. This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate. All the formulas have been checked and many corrections made. A complete bibliographical search has been conducted to present the references in modern form for ease of use. A new foreword by Professor S.J. Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity. A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text.

Page 130 of 1304