# American Institute of Mathematical Sciences

September  2019, 18(5): 2693-2715. doi: 10.3934/cpaa.2019120

## Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents

 1 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China 2 Department of Mathematics, Tsinghua University, Beijing, 100084, China

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

Received  September 2018 Revised  September 2018 Published  April 2019

Fund Project: The second author is supported by Supported by National Science Foundation of China (11571040, 11771235, 11331010).
The third author is supported by National Science Foundation of China (11571040)

This paper is concerned with the critical quasilinear Schrödinger systems in
 ${\Bbb R}^N:$
 $\left\{\begin{array}{ll}-\Delta w+(\lambda a(x)+1)w-(\Delta|w|^2)w = \frac{p}{p+q}|w|^{p-2}w|z|^q+\frac{\alpha}{\alpha+\beta}|w|^{\alpha-2}w|z|^\beta\\\ -\Delta z+(\lambda b(x)+1)z-(\Delta|z|^2)z = \frac{q}{p+q}|w|^p|z|^{q-2}z+\frac{\beta}{\alpha+\beta}|w|^\alpha|z|^{\beta-2}z, \ \end{array}\right.$
where
 $\lambda>0$
is a parameter,
 $p>2, q>2, \alpha>2, \beta>2,$
 $2\cdot(2^*-1) < p+q<2\cdot2^*$
and
 $\alpha+ \beta = 2\cdot2^*.$
By using variational method, we prove the existence of positive ground state solutions which localize near the set
 $\Omega = int \left\{a^{-1}(0)\right\}\cap int \left\{b^{-1}(0)\right\}$
for
 $\lambda$
large enough.
Citation: Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
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