Semiclassical states for fractional Choquard equations with critical growth
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January 2019, 18(1): 493-517. doi: 10.3934/cpaa.2019025
Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity
Jianhua Chen 1, , Xianhua Tang 2,, and Bitao Cheng 3,
Department of Mathematics, Nanchang University, Nanchang, 330031 Jiangxi, China
School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China
School of Mathematics and Statistics, Qujing Normal University, Qujing, 655011 Yunnan, China
Received March 2018 Revised May 2018 Published August 2018
Fund Project: This work was supported by National Natural Science Foundation of China (Grant No. 11461043, 11571370 and 11601525), and supported partly by the Provincial Natural Science Foundation of Jiangxi, China (20161BAB201009) and the Outstanding Youth Scientist Foundation Plan of Jiangxi (20171BCB23004), Yunnan Local Colleges Applied Basic Research Projects (2017FH001-011) and Hunan Provincial Innovation Foundation For Postgraduate (Grant No. CX2016B037).
In this paper, we study the following quasilinear Schrödinger equation
$\begin{equation*}-Δ u+V(x)u-Δ(u^2)u = g(u),\,\, x∈\mathbb{R}^N,\end{equation*}$
$ N>4, 2^* = \frac{2N}{N-2}, V: \mathbb{R}^N \to \mathbb{R}$
satisfies suitable assumptions. Unlike
$ g∈ \mathcal{C}^1(\mathbb{R},\mathbb{R})$
, we only need to assume that
$ g∈ \mathcal{C}(\mathbb{R},\mathbb{R})$
. By using a change of variable, we obtain the existence of ground state solutions with general critical growth. Our results extend some known results.
Keywords: Quasilinear Schrödinger equation, ground state solutions, Pohozaev identity, critical growth.
Mathematics Subject Classification: 35J50, 35J60.
Citation: Jianhua Chen, Xianhua Tang, Bitao Cheng. Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 493-517. doi: 10.3934/cpaa.2019025
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Jianhua Chen Xianhua Tang Bitao Cheng