Ground states solution of Nehari-Poho\v{z}aev type for periodic quasilinear Schr\"{o}dinger system

arXiv2305.14911MaRDI QIDQ6437770

Qian Zhang, Jianqing Chen

Publication date: 24 May 2023

Abstract: This paper is concerned with a quasilinear Schr"{o}dinger system in $mathbb{R}^N$ ${\displaystyle }$left{aligned &-Delta u+A(x)u-frac{1}{2} riangle(u^{2})u=frac{2alpha}{alpha+�eta}|u|^{alpha-2}u|v|^{�eta},\ &-Delta v+B(x)v-frac{1}{2} riangle(v^{2})v=frac{2�eta}{alpha+�eta}|u|^{alpha}|v|^{�eta-2}v,\ & u(x) o 0 hbox{and}quad v(x) o 0 hbox{as} |x| o infty,endaligned ight. ${\displaystyle }$ where $$ and $$ ($Ngeq3$). $A(x)$ and $B(x)$ are two periodic functions. By minimization under a convenient constraint and concentration-compactness lemma, we prove the existence of ground states solution. Our result covers the case of $$ which seems to be the first result for coupled quasilinear Schr"{o}dinger system in the periodic situation.

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