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

Fengshuang Gao; Yuxia Guo

doi:10.3934/dcds.2020239

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2020, Volume 40, Issue 9: 5591-5616. Doi: 10.3934/dcds.2020239

This issue Previous Article Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach Next Article

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

Fengshuang Gao and
Yuxia Guo^,

Department of Mathematical Science, Tsinghua University, Beijing, China

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

Received: January 31, 2020

Published: June 2020

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.

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

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