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Critical Hardy-Sobolev Exponents Problem with Grushin Operator and Hardy-Type Singularity Terms

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Abstract

Let \(\varOmega \) be a open set of \(\mathbb{R}^{N}\), \(0\in \varOmega \), and we are concerned with the following singular degenerate elliptic equations:

$$ -\mathcal{L}_{\alpha }u-\mu \frac{ \psi (z)^{2}u}{d(z)^{2}} = \frac{\psi (z)^{s}|u|^{2^{*}(s)-2}u}{d(z)^{s}}+\lambda f(z,u)\quad \text{in}\ \varOmega , $$

where \(\mathcal{L}_{\alpha }=\Delta _{x}+(\alpha +1)^{2}|x|^{2\alpha }\Delta _{y}\) is the Grushin operator with \(\alpha >0\), \(0< s<2\), \(d(z)\) is the natural gauge associated with \(\mathcal{L}_{\alpha }\), \(\psi (z)=|\nabla _{\alpha }d(z)|\), and \(f\) is a continuous function on \(\varOmega \) and satisfies suitable hypotheses. By means of variational methods and analytic techniques, a nonnegative nontrivial solution to this problem is obtained.

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Acknowledgements

The second author is supported by the NSFC (11761049).

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  1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, 330022, China

    Jinguo Zhang & Dengyun Yang

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  1. Jinguo Zhang
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  2. Dengyun Yang
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Correspondence to Jinguo Zhang.

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Zhang, J., Yang, D. Critical Hardy-Sobolev Exponents Problem with Grushin Operator and Hardy-Type Singularity Terms. Acta Appl Math 172, 4 (2021). https://doi.org/10.1007/s10440-021-00399-1

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