Abstract:

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 $-(\Delta_{x}+|x|^{2\alpha}\Delta_{y}) u-\mu\frac{\psi^{2}u}{d (z)^{2}}=\frac{\psi^{s}|u|^{2^*(s)-2}u}{d (z)^{s}},\, \, z=(x,y)\in\mathbb{R}^{m}\times\mathbb{R}^{n},$ 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