Infinitely many radial  solutions to  elliptic systemsinvolving  critical  exponents

Yinbin Deng; Shuangjie Peng; Li Wang

doi:10.3934/dcds.2014.34.461

Article Contents

2014, Volume 34, Issue 2: 461-475. Doi: 10.3934/dcds.2014.34.461

This issue Previous Article Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$ Next Article Variational discretization for rotating stratified fluids

Infinitely many radial solutions to elliptic systems involving critical exponents

1.
Department of Mathematics, Huazhong Normal University, Wuhan 430079
2.
Department of Mathematics, Central China Normal University, Wuhan 430079
3.
School of Basic Science, East China Jiaotong University, Nanchang 330013

Received: April 30, 2012

Revised: May 31, 2013

Published: January 2014

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Abstract

In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth $$ \left\lbrace\begin{array}{l} -\Delta u=|u|^{2^*-2}u + \frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\ -\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v + \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\ u = v = 0, \ \ &x \in \partial B, \end{array}\right. $$ where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.

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Mathematics Subject Classification: 35B33, 35J60.

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References

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