Asymmetric solutions of elliptic partial differential equations

2021 Fiscal Year Final Research Report

• Project/Area Number: 16K05236
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Saga University
• Principal Investigator: Kajikiya Ryuji 佐賀大学, 理工学部, 教授 (10183261)
• Project Period (FY): 2016-04-01 – 2022-03-31
• Keywords: 楕円型偏微分方程式 / 変分法 / 境界値問題 / 対称解

Outline of Final Research Achievements

We study the p-Laplace equation on the Dirichlet boundary condition. Under a very weak condition on the nonlinear term, we prove the existence of a sequence of solutions which converges to the zero solution. For some elliptic equations with parameters, we decide whether the zero solution is an accumulation point or an isolated point in the set of all solutions.
We study the p-Laplace Emden-Fowler equation with a radial and sign-changing weight in the unit ball under the Dirichlet boundary condition. We show that if the weight function is negative in the unit ball except for a small neighborhood of the boundary and positive at somewhere in this neighborhood, then no least energy solution is radially symmetric. Here a least energy solution is defined by the minimizer of the Rayleigh quotient.

Free Research Field

非線形楕円型偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

この研究は, 数学や物理学, 工学などにおいて非常に重要な偏微分方程式である楕円型偏微分方程式の解の性質を調べるものである. 解を無限に多くもつ楕円型偏微分方程式の研究は, 今までにも行われてきた. 本研究では, 極めて弱い仮定のもとに解の無限個の存在を証明している.
また, 方程式が対称性をもつにもかかわらずに, 解が非対称になることが起きる. このような対称性の崩れは, 解の分岐理論にも密接に関係していて, 非対称解の研究は, 楕円型偏微分方程式論において極めて重要である.