%0 Journal Article
%T 具有L2-约束的非线性Choquard方程的多解性<br>Multiple Solutions for Nonlinear Choquard Equation with L2-Constraint
%A 彭玉碧
%J Pure Mathematics
%P 65-78
%@ 2160-7605
%D 2024
%I Hans Publishing
%R 10.12677/PM.2024.141008
%X 本文考虑如下非线性Choquard方程<img src=\"Edit_7536b3ab-5e25-47e2-b026-f095d7eebba8.png\" alt=\"\" />其中a,b &gt; 0 ，α∈(0,3)，<img src=\"Edit_ccefeebd-a0c2-492f-93b5-bdb2a1e6aacc.png\" alt=\"\" />是Riesz位势。g(ξ)∈C(？, ？)满足Berestycki-Lions条件且其为奇或偶的。μ∈？是Lagrange乘子。Wu证明了(1)关于(u,κ)等同于如下系统：<img src=\"Edit_4da9b796-1d69-4a15-9dbf-e5f22203baf5.png\" alt=\"\" />在Palais-Smale-Pohozaev条件下，发展新的形变理论，使之在L<sup>2</sup>-约束问题中能应用极大极小理论并且证明该系统存在无穷多解，因此可证非线性Choquard方程也存在无穷多解。本文处理L<sup>2</sup>-约束问题，即∫<sub>？<sup>3</sup></sub>|u|<sup>2</sup>dx=m。<br />
In this paper, we consider the following nonlinear Choquard equation <img src=\"Edit_cead92cf-104e-44ea-b888-c8ab62cb038e.png\" alt=\"\" />wherea,b &gt; 0 ，α∈(0,3)，<img src=\"Edit_0e58d87c-b65d-4044-b814-5ac55e0332fe.png\" alt=\"\" />is a Riesz potential. g(ξ)∈C(？, ？) satisfies Berestycki-Lions condition and it is odd or even. μ∈？ is a Lagrange multiplier. Wu proved that (1) is equivalent to the following system with respect to (u,κ): <img src=\"Edit_8f29f94b-2ed7-4fb9-a3bc-360698b3d163.png\" alt=\"\" />We develop a new deformation argument under Palais-Smale-Pohozaev condition. It enables us to apply minimax argument for L<sup>2</sup>-constraint problem and we can prove the system exists infinitely many solutions, so we also prove Nonlinear Choquard Equation exists infinitely many solutions. In this paper, we deal with L<sup>2</sup>-constraint problem, i.e. ∫<sub>？<sup>3</sup></sub>|u|<sup>2</sup>dx=m.
%K 非线性Choquard方程，Riesz位势，多维奇路径，Berestycki-Lions条件，L<sup>2</sup>-约束问题<br>Nonlinear Choquard Equation
%K Riesz Potential
%K Multidimensional Odd Paths
%K Berestycki-Lions Condition
%K L<sup>2</sup>-Constraint Problem
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=79463