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

Yongpeng Chen; Yuxia Guo; Zhongwei Tang

doi:10.3934/cpaa.2019120

Article Contents

2019, Volume 18, Issue 5: 2693-2715. Doi: 10.3934/cpaa.2019120

This issue Previous Article Faber-Krahn and Lieb-type inequalities for the composite membrane problem Next Article Scattering in the weighted

$ L^2 $

-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

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

Received: August 31, 2018

Revised: August 31, 2018

Published: March 2019

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J655.

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