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
摘要: 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。
Abstract: 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.
文章引用：彭玉碧. 具有L2-约束的非线性Choquard方程的多解性[J]. 理论数学, 2024, 14(1): 65-78. https://doi.org/10.12677/PM.2024.141008

参考文献

[1] Pekar, S.I. (1954) Untersuchungen über die Elektronentheorie der Kristalle. Akademie Verlag, Berlin. [Google Scholar] [CrossRef] 
[2] Lieb, E.H. (1977) Existence and Uniqueness of the Minimizing Solution of Choquard’s Nonlinear Equation. Studies in Applied Mathematics, 57, 93-105. [Google Scholar] [CrossRef] 
[3] Hirata, J., Ikoma, N. and Tanaka, K. (2010) Nonlinear Scalar Field Equations in : Mountain Pass and Symmetric Mountain Pass Approaches. Topological Methods in Nonlinear Analysis, 35, 253-276.
[4] Hirata, J. and Tanaka, K. (2019) Nonlinear Scalar Field Equations with L2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches. Advanced Nonlinear Studies, 19, 263-290. [Google Scholar] [CrossRef] 
[5] Kirchhoff, G. (1883) Mechanik. Teubner, Leipzig.
[6] Arosio, A. and Panizzi, S. (1996) On the Well-Posedness of the Kirchhoff String. American Mathematical Society, 348, 305-330. [Google Scholar] [CrossRef] 
[7] Cavalcanti, M.M., Cavalcanti, V.N.D. and Soriano, J.A. (2001) Global Existence and Uniform Decay Rates for the Kirchhoff-Carrier Equation with Nonlinear Dissipation. Advances in Difference Equations, 6, 701-730. [Google Scholar] [CrossRef] 
[8] D’Ancona, P. and Spagnolo, S. (1992) Global Solvability for the Degenerate Kirchhoff Equation with Real Analytic Data. Inventiones Mathematicae, 108, 247-262. [Google Scholar] [CrossRef] 
[9] He, X. and Zou, W. (2010) Multiplicity of Solutions for a Class of Kirchhoff Type Problems. Acta Mathematicae Applicatae Sinica, 26, 387-394. [Google Scholar] [CrossRef] 
[10] Cingolani, S., Gallo, M. and Tananka, K. (2022) Multiple Solu-tions for the Nonlinear Choquard Equation with Even or Odd Nonlinearities. Calculus of Variations and Partial Dif-ferential Equations, 68, 1-34. [Google Scholar] [CrossRef] 
[11] Berestycki, H. and Lions, P.L. (1983) Nonlinear Scalar Field Equations II: Existence of Infinitely Many Solutions. Archive for Rational Mechanics and Analysis, 82, 347-375. [Google Scholar] [CrossRef] 
[12] Wu, K. and Zhou, F. (2019) Nodal Solutions for a Kirchhoff Type Problem in . Applied Mathematics Letters, 88, 58-63. [Google Scholar] [CrossRef] 
[13] Lieb, E.H. (1983) Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities. Annals of Mathematics, 118, 349-374. [Google Scholar] [CrossRef] 
[14] Cingolani, S., Gallo, M. and Tanaka, K. (2021) Normalized Solu-tions for Fractional Nonlinear Scalar Field Equation via Lagrangian Formulation. Nonlinearity, 34, 4017-4056. [Google Scholar] [CrossRef] 

为你推荐