Multiple solutions for a critical quasilinear equation with Hardy potential

Fengshuang Gao; Yuxia Guo

doi:10.3934/dcdss.2019128

Article Contents

2019, Volume 12, Issue 7: 1977-2003. Doi: 10.3934/dcdss.2019128

This issue Previous Article Branching and bifurcation Next Article Global symmetry-breaking bifurcations of critical orbits of invariant functionals

Multiple solutions for a critical quasilinear equation with Hardy potential

Fengshuang Gao^1, and
Yuxia Guo^1, ,

Department of Mathematical Science, Tsinghua University, Beijing, China

* Corresponding author
* Corresponding author

Received: November 30, 2017

Revised: March 31, 2018

Published: June 2019

The authors are supported by NSFC grant 11771235, 11331010, 11571040.

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Abstract

In this paper, we investigate the following quasilinear equation involving a Hardy potential:

$\begin{array}{l}\left\{ {\begin{array}{*{20}{c}}{ - \sum\limits_{i,j = 1}^N {{D_j}} ({a_{ij}}(u){D_i}u) + \frac{1}{2}\sum\limits_{i,j = 1}^N {{{a'}_{ij}}} (u){D_i}u{D_j}u - \frac{\mu }{{|x{|^2}}}u = au + |u{|^{{2^ * } - 2}}u}&{{\rm{in}}\;\Omega ,}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{u = 0}&{{\rm{on}}\;\partial \Omega ,}\end{array}} \right.\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {\rm{P}} \right)\end{array}$

where $ 2^ * = \frac{2N}{N-2} $ is the Sobolev critical exponent for the embedding of $ H_0^1(\Omega) $ into $ L^p(\Omega) $, $ a>0 $ is a constant and $ \Omega\subset \mathbb{R}^N $ is an open bounded domain which contains the origin. We will prove that under some suitable assumptions on $ a_{ij} $, when $ N\geq 7 $ and $ \mu\in[0,\mu^*) $ for some constant $ \mu^* $, problem (P) admits an unbounded sequence of solutions. To achieve this goal, we perform the subcritical approximation and the regularization perturbation.

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Mathematics Subject Classification: 35B45, 35J25.

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