A perturbed problem of elliptic system with critical exponent

ABSTRACT

This paper deals with the following perturbed problem with critical exponent (1) $\left\{\begin{array}{ll}-\mathrm{\Delta }u=u^{2^{\ast }-1}+\frac{\alpha }{2^{\ast }}{u}^{\alpha -1}{v}^{\text{β}}+ϵK\left(x\right)u^p,& x\in {\mathbb{R}}^N,\\ -\mathrm{\Delta }v=v^{2^{\ast }-1}+\frac{\text{β}}{2^{\ast }}{u}^{\alpha }{v}^{\text{β}-1}+ϵQ\left(x\right)v^q,& x\in {\mathbb{R}}^N,\\ u>0,v>0,& x\in {\mathbb{R}}^N,\end{array}\right.$(1) where $1\le p,q<2^{\ast }-1$, $\alpha +\text{β}=2^{\ast }:=\frac{2N}{N-2}$, $\alpha ,\text{β}\ge 2$, N = 3, 4 and ε is a parameter. Using a perturbation argument and Lyapunov–Schmidt reduction method, we obtain the existence of positive solutions to problem (1) and the asymptotic property of the solutions.

Keywords:

• Perturbation argument
• Lyapunov–Schmidt reduction method
• critical exponent

2010 Mathematics Subject Classifications:

• 35J47
• 35J50

Acknowledgments

Both authors would like to thank Professor Shuangjie Peng very much for stimulating discussions and helpful suggestions on the present paper. The research of Qi Li was supported by the excellent doctorial dissertation cultivation grant ( No. 2019YBZZ057) from Central China Normal University.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of Q. Li was supported by the excellent doctorial dissertation cultivation grant [number 2019YBZZ057] from Central China Normal University.