Multiple entire solutions of fractional Laplacian Schrödinger equations

Jian Wang; Zhuoran Du; Jian Wang; Zhuoran Du

doi:10.3934/math.2021494

AIMS Mathematics

2021, Volume 6, Issue 8: 8509-8524. doi: 10.3934/math.2021494

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Research article

Multiple entire solutions of fractional Laplacian Schrödinger equations

Jian Wang ,
Zhuoran Du ^,

School of Mathematics, Hunan University, Changsha 410082, China

Received: 19 February 2021 Accepted: 19 May 2021 Published: 04 June 2021
MSC : 35A01, 35B08, 35J61

We consider the semi-linear fractional Schrödinger equation

$ \begin{align*} \begin{cases} (-\Delta)^s u+V(x)u = f(x,u), \; \; \; x\in \mathbb{R}^N ,\\ u\in H^{s}(\mathbb {R}^N) , \\ \end{cases} \end{align*} $

where both $ V(x) $ and $ f(x, u) $ are periodic in $ x $, $ 0 $ belongs to a spectral gap of the operator $ (-\Delta)^s+V $ and $ f(x, u) $ is subcritical in $ u $. We obtain the existence of nontrivial solutions by using a generalized linking theorem, and based on this existence we further establish infinitely many geometrically distinct solutions. We weaken the super-quadratic condition of $ f $, which is usually assumed even in the standard Laplacian case so as to obtain the existence of solutions.
- fractional Schrödinger equation,
- Cerami sequence,
- infinitely many geometrically distinct solutions
Citation: Jian Wang, Zhuoran Du. Multiple entire solutions of fractional Laplacian Schrödinger equations[J]. AIMS Mathematics, 2021, 6(8): 8509-8524. doi: 10.3934/math.2021494

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Abstract

We consider the semi-linear fractional Schrödinger equation

$ \begin{align*} \begin{cases} (-\Delta)^s u+V(x)u = f(x,u), \; \; \; x\in \mathbb{R}^N ,\\ u\in H^{s}(\mathbb {R}^N) , \\ \end{cases} \end{align*} $

where both $ V(x) $ and $ f(x, u) $ are periodic in $ x $, $ 0 $ belongs to a spectral gap of the operator $ (-\Delta)^s+V $ and $ f(x, u) $ is subcritical in $ u $. We obtain the existence of nontrivial solutions by using a generalized linking theorem, and based on this existence we further establish infinitely many geometrically distinct solutions. We weaken the super-quadratic condition of $ f $, which is usually assumed even in the standard Laplacian case so as to obtain the existence of solutions.

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