尖锐的Gagliardo-Nirenberg不等式及其在非齐次非线性分数型问题中的应用

IF 1.3 4区 数学 Q1 MATHEMATICS
D. Bhimani, H. Hajaiej, S. Haque, Tingjian Luo
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引用次数: 7

摘要

本文的目的有三个。首先,我们建立了一个包含分数范数和非齐次非线性的最优常数Gagliardo-Nirenberg不等式。其次,作为该不等式的应用,我们研究了具有混合分数阶拉普拉斯方程和非齐次非线性的非线性Schrödinger方程(NLS)的基态驻波,并考虑了具有规定质量的基态解存在性的最小化问题。特别地,利用Gagliardo-Nirenberg不等式及其最优常数,给出了存在性结果的充要条件。最后,我们建立了具有混合分数阶拉普拉斯算子和非齐次非线性的NLS的局部适定性理论。在此过程中,我们证明了Lorentz空间中的Strichartz估计,这可能是一个独立的兴趣。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A sharp Gagliardo-Nirenberg inequality and its application to fractional problems with inhomogeneous nonlinearity
The purpose of this paper is threefold. Firstly, we establish a Gagliardo-Nirenberg inequality with optimal constant, which involves a fractional norm and an inhomogeneous nonlinearity. Secondly, as an application of this inequality, we study ground state standing waves to a nonlinear Schrödinger equation (NLS) with a mixed fractional Laplacians and a inhomogeneous nonlinearity, and consider a minimization problem which gives the existence of ground state solutions with prescribed mass. In particular, by making use of this Gagliardo-Nirenberg inequality and its optimal constant, we give a sufficient and necessary condition for the existence results. Finally, we develop local wellposedness theory for NLS with a mixed fractional Laplacians and a inhomogeneous nonlinearity. In the process, we prove Strichartz estimates in Lorentz spaces which may be of independent interest.
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来源期刊
Evolution Equations and Control Theory
Evolution Equations and Control Theory MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
3.10
自引率
6.70%
发文量
5
期刊介绍: EECT is primarily devoted to papers on analysis and control of infinite dimensional systems with emphasis on applications to PDE''s and FDEs. Topics include: * Modeling of physical systems as infinite-dimensional processes * Direct problems such as existence, regularity and well-posedness * Stability, long-time behavior and associated dynamical attractors * Indirect problems such as exact controllability, reachability theory and inverse problems * Optimization - including shape optimization - optimal control, game theory and calculus of variations * Well-posedness, stability and control of coupled systems with an interface. Free boundary problems and problems with moving interface(s) * Applications of the theory to physics, chemistry, engineering, economics, medicine and biology
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