Acta mathematica scientia,Series A ›› 2021, Vol. 41 ›› Issue (4): 997-1012.

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Existence and Asymptotic Behavior of Solution for a Degenerate Elliptic Equation Involving Grushin-Type Operator and Critical Sobolev-Hardy Exponents

Jinguo Zhang*(),Dengyun Yang()   

  1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022
  • Received:2020-05-13 Online:2021-08-26 Published:2021-08-09
  • Contact: Jinguo Zhang;
  • Supported by:
    the NSFC(11761049)


In this paper, we study the existence and asymptotic behavior of solutions for a class of degenerate elliptic equation involving Grushin-type operator and Hardy potentials where $-(\Delta_{x}+|x|^{2\alpha}\Delta_{y}) $ is the Grushin-type operator, $\alpha>0, 2^*(s)=\frac{2(Q-s)}{Q-2} $ is the critical Sobolev-Hardy exponent and $Q=m+(\alpha+1)n $ is the homogenous dimension for Grushin operator. If $0 \leq \mu<(\frac{Q-2}{2})^{2}, 0 < s <2$, we will prove the existence of nontrivial, nonnegative solutions for this degenerate problem, and give the asymptotic behavior of solutions, at the singularity and at infinity.

Key words: Grushin-type operator, Moser iteration, Asymptotic behavior, Critical Sobolev-Hardy exponents

CLC Number: 

  • O175.29