A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY

doi:10.1007/s10473-022-0205-x

Abstract

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

Chunyu LEI, Jiafeng LIAO, Changmu CHU, Hongmin SUO. A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY[J].Acta mathematica scientia,Series B, 2022, 42(2): 502-510.

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