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Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $ \mathbb{R}^N $

  • * Corresponding author: yguo@mail.tsinghua.edu.cn

    * Corresponding author: yguo@mail.tsinghua.edu.cn

The second author is supported by NSFC grant 11771235.

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  • In this paper, we consider the following critical quasilinear equation with Hardy potential:

    $ \begin{equation} \!\!\!\!\!\left\{ \begin{array}{ll} \!\!\!\!\!-\!\!\sum\limits_{i,j = 1}^N\!\!\!D_j(a_{ij}(u)D_iu)\!+\!\frac{1}{2}\!\!\!\sum\limits_{i,j = 1}^N\!\!\!\!a_{ij}'(u)D_iuD_ju\!+\!a(x)u\! = \!\nu |u|^{q-2}u\!+\!\frac{\mu u}{|x|^2}\!+\!|u|^{2^*-2}u,\hbox{in } \mathbb{R}^N,\\ u(x)\to0\,\,\hbox{as } |x|\,\,\to\infty, \end{array}\right. \;\;\;\;\;(1)\end{equation} $

    where $ a_{ij}(u)\!\in \!C^1\!(\mathbb{R},\mathbb{R}) $, $ \nu\!>\!0 $, $ 0\!\leq\!\mu\!<\!\alpha\bar{\mu} $, and $ \max\!\left\{\!\frac{\alpha\bar{\mu}\gamma}{\alpha\bar{\mu}-\mu}\!+\!2,2^*\!\!-\!\frac{2}{N-2}\sqrt{\!\bar{\mu}\!-\!\frac{\mu}{\alpha}}\right\} \!\!<q<2^* $, $ \alpha, \gamma>0 $, $ \bar{\mu} = \frac{(N-2)^2}{4} $, $ 2^\ast = \frac{2N}{N-2} $ is the Sobolev critical exponent. And $ a(x) $ is a finite, positive potential function satisfying suitable decay assumptions. By using truncation method combining with the regularization approximation approach and compactness arguments, we prove the existence of infinitely many solutions for this equation.

    Mathematics Subject Classification: 35A01, 35J20, 35J62.


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