Multi-peak semiclassical bound states  for Fractional Schrödinger Equations with fast decaying potentials

Xiaoming An; Shuangjie Peng; Xiaoming An; Shuangjie Peng

doi:10.3934/era.2022031

Electronic Research Archive

2022, Volume 30, Issue 2: 585-614. doi: 10.3934/era.2022031

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Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials

Xiaoming An ¹,
Shuangjie Peng ^{2
,
,}

1.
School of Mathematics and Statistics & Guizhou University of Finance and Economics, Guiyang, 550025, China
2.
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

Received: 21 September 2021 Revised: 25 November 2021 Accepted: 25 November 2021 Published: 15 February 2022

We study the following fractional Schrödinger equation

$ \begin{equation*} \label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} $

where $ s\in(0,1) $. Under some conditions on $ f(u) $, we show that the problem has a family of solutions concentrating at any finite given local minima of $ V $ provided that $ V\in C( \mathbb{R}^N,[0,+\infty)) $. All decay rates of $ V $ are admissible. Especially, $ V $ can be compactly supported. Different from the local case $ s = 1 $ or the case of single-peak solutions, the nonlocal effect of the operator $ (-\Delta)^s $ makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method.
- variational method,
- fractional Schrödinger,
- multi-peak,
- compactly supported,
- penalized technique
Citation: Xiaoming An, Shuangjie Peng. Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials[J]. Electronic Research Archive, 2022, 30(2): 585-614. doi: 10.3934/era.2022031

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Abstract

We study the following fractional Schrödinger equation

$ \begin{equation*} \label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} $

where $ s\in(0,1) $. Under some conditions on $ f(u) $, we show that the problem has a family of solutions concentrating at any finite given local minima of $ V $ provided that $ V\in C( \mathbb{R}^N,[0,+\infty)) $. All decay rates of $ V $ are admissible. Especially, $ V $ can be compactly supported. Different from the local case $ s = 1 $ or the case of single-peak solutions, the nonlocal effect of the operator $ (-\Delta)^s $ makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method.

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