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›› 2011, Vol. 28 ›› Issue (5): 583-590.DOI: 10.7523/j.issn.2095-6134.2011.5.003

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Some remarks on a quasilinear elliptic equation with critical exponent

LIU Xing, SUN Yi-Jing   

  1. School of Mathematics, Graduate University, Chinese Academy of Sciences, Beijing 100049, China
  • Received:2010-10-09 Revised:2010-11-08 Online:2011-09-15

Abstract:

We investigate the following quasilinear elliptic equation:

Δpu + uq + λup*-1 =0 , uW1,p0(Ω), (1λ)


where Ω is a bounded domain in RN with smooth boundary, Δpu=div(| u |p-2 u), N≥3, 2≤p < N, 0<q<1, and p*= (Np)/(N-p). By using variational methods, we obtain a lower bound of the extremal value λ*(Ω,p,q) for equation (1λ), which can be explicitly calculated.

 

 

Key words: quasilinear elliptic equation, critical exponent, Ekeland’s variational principle, extremal value

CLC Number: