Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term

Chao Ji

doi:10.3934/dcdsb.2019131

Article Contents

2019, Volume 24, Issue 11: 6071-6089. Doi: 10.3934/dcdsb.2019131

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Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term

Chao Ji^1,2, ,

1.
Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
2.
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

^* Corresponding author: Chao Ji
^* Corresponding author: Chao Ji

Received: October 2017

Revised: August 2018

Early access: May 2019

Published: August 2019

C. Ji was supported by Shanghai Natural Science Foundation(18ZR1409100), NSFC (grant No. 11301181, 11771324) and China Postdoctoral Science Foundation funded project

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Abstract

In this paper we are concerned with the fractional Schrödinger equation $ (-\Delta)^{\alpha} u+V(x)u = f(x, u) $, $ x\in {{\mathbb{R}}^{N}} $, where $ f $ is superlinear, subcritical growth and $ u\mapsto\frac{f(x, u)}{\vert u\vert} $ is nondecreasing. When $ V $ and $ f $ are periodic in $ x_{1},\ldots, x_{N} $, we show the existence of ground states and the infinitely many solutions if $ f $ is odd in $ u $. When $ V $ is coercive or $ V $ has a bounded potential well and $ f(x, u) = f(u) $, the ground states are obtained. When $ V $ and $ f $ are asymptotically periodic in $ x $, we also obtain the ground states solutions. In the previous research, $ u\mapsto\frac{f(x, u)}{\vert u\vert} $ was assumed to be strictly increasing, due to this small change, we are forced to go beyond methods of smooth analysis.

Keywords:

Fractional logarithmic Schrödinger equation,
periodic potential,
coercive potential,
bounded potential,
nonsmooth critical point theory.

Mathematics Subject Classification: Primary: 35J60, 35R11; Secondary: 47J30.

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