Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (2): 502-510.doi: 10.1007/s10473-022-0205-x

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A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY

Chunyu LEI1, Jiafeng LIAO2, Changmu CHU3, Hongmin SUO3   

  1. 1. School of Sciences, GuiZhou Minzu University, Guiyang 550025, China;
    2. College of Mathematics Education, China West Normal University, Nanchong 637002, China;
    3. School of Sciences, GuiZhou Minzu University, Guiyang 550025, China
  • Received:2020-03-23 Revised:2021-03-17 Online:2022-04-25 Published:2022-04-22
  • Supported by:
    The first author is supported by Natural Science Foundation of Guizhou Minzu University ([2018]5773-YB03); the second author is supported by Fundamental Research Funds of China West Normal University (18B015) and Innovative Research Team of China West Normal University (CXTD2018-8); the third author is supported by National Natural Science Foundation of China (11861021); the fourth author is supported by National Natural Science Foundation of China (11661021).

Abstract: We consider the logarithmic elliptic equation with singular nonlinearity \begin{equation*} \begin{cases} \Delta u+u\log u^2 +\frac{\lambda}{u^\gamma}=0, &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ ($N\geq3$) is a bounded domain with a smooth boundary, $0<\gamma<1$ and $\lambda$ is a positive constant. By using a variational method and the critical point theory for a nonsmooth functional, we obtain the existence of two positive solutions. This result generalizes and improves upon recent results in the literature.

Key words: Logarithmic elliptic equation, singular nonlinearity, positive solutions, variational method

CLC Number: 

  • 35A15
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