Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities
Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities
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DOI：
10.1007/s00526-021-02182-4
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发表时间：
2022-02
影响因子：
2.1
通讯作者：
S. Cingolani;Marco Gallo;Kazunaga Tanaka
中科院分区：
文献类型：
--
作者：
S. Cingolani;Marco Gallo;Kazunaga Tanaka
我们证明了非线性Choquard方程的无穷多解的存在性：非线性Choquard方程\Documentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setlong{\oddsidemarin}{-69pt}\Begin{Document}$$\Begin{Align}-{\varDelta}u+\Mu u=(i_\α*F(U))f(U)\quad\Hbox{in}\{\mathbb{R}}^N，其中，，是Riesz势，F是几乎最优的次临界非线性，假设是奇数或偶数。我们分析了两种情况：是固定的正常数或未知的，并且解的-范数是规定的，即。由于非局域性的存在阻碍了应用由Berestycki和Lions(Arch Ratio Mech Anal 82(4)：347-375,1983)提出的经典方法，我们实现了一种新的多维奇路径的构造，其中Riesz势的一些估计起着至关重要的作用，并且我们找到了他们的多重性结果的非局域对应。特别地，我们推广了Moroz和Van Schaftingen的存在性结果(Trans am Math Soc 367(9)：6557-6579,2015)。
We prove existence of infinitely many solutionsfor the nonlinear Choquard equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - {\varDelta } u + \mu u =(I_\alpha *F(u)) f(u) \quad \hbox {in}\ {\mathbb {R}}^N, \end{aligned}$$\end{document}where,,,is the Riesz potential, andFis an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases:is a fixed positive constant oris unknown and the-norm of the solution is prescribed, i.e.. Since the presence of the nonlocality prevents to apply the classical approach, introduced by Berestycki and Lions (Arch Ration Mech Anal 82(4):347–375, 1983), we implement a new construction of multidimensional odd paths, where some estimates for the Riesz potential play an essential role, and we find a nonlocal counterpart of their multiplicity results. In particular we extend the existence results due to Moroz and Van Schaftingen (Trans Am Math Soc 367(9):6557–6579, 2015).