Generalized quasilinear Schrödinger equations with concave functions $ l(s^2) $

Yongkuan Cheng; Yaotian Shen

doi:10.3934/dcds.2019056

Article Contents

2019, Volume 39, Issue 3: 1311-1343. Doi: 10.3934/dcds.2019056

This issue Previous Article Direct methods on fractional equations Next Article Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians

Generalized quasilinear Schrödinger equations with concave functions $ l(s^2) $

Yongkuan Cheng and
Yaotian Shen^,

School of Mathematics, South China University of Technology, Guangzhou 510640, China

* Corresponding author
* Corresponding author

Received: December 31, 2017

Published: December 2018

Supported by NSFC (No.11371146), the Fundamental Research Funds for the Central Universities (No.2015zz133)

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Abstract

We establish the existence of nontrivial solutions for the following quasilinear Schrödinger equation with subcritical or critical growth:

$ \begin{equation*}-Δ u+W(x)u^{2α-1}-ul'(u^2)Δ l(u^2) = f(u) \ \mbox{or}\ h(u)+u^{2^*-1},\ x∈\mathbb{R}^N,\end{equation*} $

where $W(x):\mathbb{R}^N \to \mathbb{R} $ is a given potential and $ l,h,f $ are real functions, $ u>0,$ $ 2^* = 2N/(N-2), $ $ N≥3 $. Our results cover physical models $ l(s) = s^{\frac{α}{2}}, $ $ \frac{1}{2}<α<1. $

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Mathematics Subject Classification: 5J20, 35J62, 35Q55.

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