Variational approaches to some class of quasilinear elliptic equations

2023 Fiscal Year Final Research Report

• Project/Area Number: 18K03362
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Shizuoka University
• Principal Investigator: Adachi Shinji 静岡大学, 工学部, 教授 (40339685)
• Co-Investigator(Kenkyū-buntansha): 柴田 将敬 名城大学, 理工学部, 准教授 (90359688) ; 渡辺 達也 京都産業大学, 理学部, 教授 (60549749)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Keywords: 準線形楕円型方程式 / 変分解析 / 正値解 / 漸近挙動

Outline of Final Research Achievements

By using variational methods, I studied the uniqueness of positive solutions and its asymptotic behavior for some class of quasilinear elliptic equation. Since this quasilinear elliptic equation has a dual variational structure, I could transform quasilinear elliptic equation into a semilinear one. As a byproduct of this approach, I also could extend previous works on the uniqueness of positive solutions for some semilinear elliptic equations of scalar field type. As for the asymptotic behavior, I found that an appropriate self-similar transformation of positive solution converges to the Talenti function, and the asymptotic profile is completely revealed.
I also clarified the existence of positive solution and its asymptotic behavior for semilinear elliptic equations that do not impose any growth condition at infinity.

Free Research Field

偏微分方程式，変分解析

Academic Significance and Societal Importance of the Research Achievements

近年臨界点理論の発展に伴い，準線形楕円型方程式への変分的手法の適用について活発に研究されるようになり，特にプラズマ物理学に端を発するシュレディンガー方程式の定在波解を与える準線形楕円型方程式の変分的研究が当該分野の主題のひとつとなっている。本研究ではこの方程式の解構造，特に正値解の一意性およびその漸近挙動，漸近的プロファイルを明らかにした。この研究成果は従来の優線形劣臨界増大度の非線形項を持つ半線形楕円型方程式に対する研究にも新たな切り口を与えた。また，数学的厳密化は物理学的応用研究の発展にも寄与できる大きな意義のある研究である。