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Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent

Volume 116 / 2016

Jia-Feng Liao, Jiu Liu, Peng Zhang, Chun-Lei Tang Annales Polonici Mathematici 116 (2016), 273-292 MSC: Primary 35J15; Secondary 35A15, 35B09. DOI: 10.4064/ap3606-10-2015 Published online: 16 March 2016

Abstract

We study the following singular elliptic equation with critical exponent $$ \begin{cases} -\varDelta u=Q(x)u^{2^{*}-1}+\lambda u^{-\gamma}&\text{in } \varOmega, \\ u \gt 0 &\text{in } \varOmega, \\ u=0 &\text{on } \partial\varOmega, \end{cases} $$ where $\varOmega\subset\mathbb{R}^{N}$ $(N\geq3)$ is a smooth bounded domain, and $\lambda \gt 0$, $\gamma\in(0,1)$ are real parameters. Under appropriate assumptions on $Q,$ by the constrained minimizer and perturbation methods, we obtain two positive solutions for all $\lambda \gt 0$ small enough.

Authors

  • Jia-Feng LiaoSchool of Mathematics and Statistics
    Southwest University
    400715 Chongqing
    People’s Republic of China
    and
    School of Mathematics
    and Computational Science
    Zunyi Normal College
    563002 Zunyi, Guizhou
    People’s Republic of China
    e-mail
  • Jiu LiuSchool of Mathematics and Statistics
    Southwest University
    400715 Chongqing
    People’s Republic of China
    e-mail
  • Peng ZhangSchool of Mathematics
    and Computational Science
    Zunyi Normal College
    563002 Zunyi, Guizhou
    People’s Republic of China
    e-mail
  • Chun-Lei TangSchool of Mathematics and Statistics
    Southwest University
    400715 Chongqing
    People’s Republic of China
    e-mail

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