具有L2-约束的非线性Choquard方程的多解性

Multiple Solutions for Nonlinear Choquard Equation with L2-Constraint

本文考虑如下非线性Choquard方程其中a,b > 0 ,α∈(0,3),是Riesz位势。g(ξ)∈C(ℝ, ℝ)满足Berestycki-Lions条件且其为奇或偶的。μ∈ℝ是Lagrange乘子。Wu证明了(1)关于(u,κ)等同于如下系统:在Palais-Smale-Pohozaev条件下,发展新的形变理论,使之在L2-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L2-约束问题,即∫ℝ3|u|2dx=m。

In this paper, we consider the following nonlinear Choquard equation wherea,b > 0 ,α∈(0,3),is a Riesz potential. g(ξ)∈C(ℝ, ℝ) satisfies Berestycki-Lions condition and it is odd or even. μ∈ℝ is a Lagrange multiplier. Wu proved that (1) is equivalent to the following system with respect to (u,κ): We develop a new deformation argument under Palais-Smale-Pohozaev condition. It enables us to apply minimax argument for L2-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 L2-constraint problem, i.e. ∫ℝ3|u|2dx=m.