Solvability and solutions' analysis of nonlinear elliptic equations from the viewpoint of eigenvalue problems

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03591
• 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: Tokyo University of Science
• Principal Investigator: Tanaka Mieko 東京理科大学, 理学部第一部数学科, 准教授 (00459728)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 非線形固有値問題 / 楕円型微分作用素 / 解の存在と非存在 / 解の符号

Outline of Final Research Achievements

(1)By generalizing the Picone inequality used in the p-Laplace equation and using a good test function, the Picone inequality can be used to prove the non-existence of positive solutions to the (p,q)Laplace equation. (2)Although two curves related to the (p,q)-Laplacian eigenvalue problem were constructed in the previous work, it’s proven that the two curves do not coincide by showing the existence of positive solutions that are different from the least energy solution. By this method, we succeeded in finding three positive solutions in a special case. (3) In the p-Laplace eigenvalue equation with a p-sublinear perturbation term, we showed the existence of two positive solutions that seem to be the bifurcation from the least energy solution even when a parameter is over the threshold.

Free Research Field

楕円型微分方程式、変分的手法

Academic Significance and Societal Importance of the Research Achievements

Picone不等式を一般化して上手いテスト関数を用いる事により適用出来る方程式の範囲を広げる事に成功した事は、色々な形への一般化Picone不等式の導出と適用方法の改良などへの促進となり今後の発展が期待される。正値解の多重存在の証明方法を構築や既存の方法の改良を行った。この手法が他の方程式などにも適用されたり、改良されて使われるようになる事が期待され、意義があると考える。