Study on nonlinear elliptic partial differential equations via variational method and perturbation methods

2019 Fiscal Year Final Research Report

• Project/Area Number: 15K17567
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Saitama University
• Principal Investigator: Sato Yohei 埼玉大学, 理工学研究科, 准教授 (00465387)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: 楕円型偏微分方程式 / 変分法 / 摂動法

Outline of Final Research Achievements

In the research period, 7 papers are published or accepted. We studied the solutions structure of system which consists of three elliptic partial differential equations with an attractive coupling and two repulsive couplings. Consequently, We obtained new results with respect to the multiple existence of positive solutions, the non-symmetricity of the solutions, and the multiple existence of sign-changing solutions. We also proved the existence of infinitely many solutions to elliptic partial differential equations whose the potential function approaches a positive constant at infinity under more general assumptions of nonlinearity. In addition, when the spatial dimension is 1, we also proved the nonexistence of nontrivial solutions. We also obtained the results with respect to the localized solutions of the nonlinear Schr\"odinger system with critical frequency for infinite attractive case.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

連立シュレディンガー方程式は、比較的最近研究が試みられるようになった方程式であるが、連立非線形シュレディンガー方程式の変分構造は単独の方程式の変分構造と似ているため、多くの研究が急速に発表されていた。しかし、単独の方程式と連立方程式の解構造の本質的な違いを抽出するような研究結果はそれほど多くはなかった。本研究は、連立方程式では、たとえ方程式の係数が空間変数に依存しなくても、複雑な解構造を持ち得ることを明らかにした。