Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $

Ziqing Yuan; Jianshe Yu

doi:10.3934/dcdss.2020281

Article Contents

2021, Volume 14, Issue 9: 3285-3303. Doi: 10.3934/dcdss.2020281

This issue Previous Article Chaotic oscillations of linear hyperbolic PDE with variable coefficients and implicit boundary conditions Next Article $ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $

Ziqing Yuan^1,2, and
Jianshe Yu^1, ,

1.
Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong, 510006, China
2.
Department of Mathematics, Shaoyang University, Shaoyang, Hunan, 422000, China

^* Corresponding author: Jianshe Yu
^* Corresponding author: Jianshe Yu

Received: June 30, 2019

Revised: September 30, 2019

Early access: March 2020

Published: September 2021

Research are supported by the National Natural Science Foundation of China (Grant No. 11901126), the Hunan Province Natural Science Foundation of China (Grant No. 2017JJ3222), and the China Postdoctoral Science Foundation funded project (Grant No. 2018M643039)

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Abstract

This work concerns with the existence and multiplicity of positive solutions for the following quasilinear Schrödinger equation

$ -\Delta u+V(x)u-\Delta(u^2)u = a(\epsilon x)g(u), \; \; \; \; x\in\mathbb R^N, $

where $ V(x)>0 $, $ u>0 $, $ a $ and $ g $ are continuous functions and $ a $ has $ m $ maximum points. With the change of variables we show that this equation has at least $ m $ nontrivial solutions by using variational methods, the Ekeland's variational principle, and some properties of the Nehari manifold. Some recent results are improved.

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Mathematics Subject Classification: 35J85; 47J30; 49J52.

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