Study of the strucure of solutions to nonlinear elliptic equations with various effects

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K03590
• 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: Keio University
• Principal Investigator: Ikoma Norihisa 慶應義塾大学, 理工学部(矢上), 准教授 (50728342)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 非自明解 / Born-Infeld方程式 / 弱解と最小化元 / 光線分 / 分数冪ラプラシアン / 安定解 / 層構造

Outline of Final Research Achievements

The aim of this research project was to investigate the structure of nontrivial solutions to elliptic equations involving singularities, nonlocalities and so on. During the period, the following results were obtained. For the Born-Infeld equation(this equation has a singularity), the regularity of minimizer as well as the relation between minimizers and weak solutions were studied. For the equation with the fractional Laplacian and the Hardy-Henon type nonlinearity (the equation has a nonlocality), the existence and nonexistence of stables solutions was proved. The layer property of the family of stable solutions were also shown. In addition to these two equations, the existence of nontrivial solutions and their properties were obtained for the equation with sublinear nonlinearities, a class of equations involving the 1 dimensional Pucci operators, the equation with large parameters and the equation with a constraint on the L^2 norm of solutions.

Free Research Field

偏微分方程式

Academic Significance and Societal Importance of the Research Achievements

Born-Infeld方程式については，これまでの先行研究とは異なる着眼点から研究を行い，得られた成果も先行研究とは大きく異なる．特に光線分を持つ弱解の存在を示し，エネルギー汎関数の最小化元だが弱解とはならない例を構成できたことも非常に意義深い．分数冪ラプラシアンを含む方程式では，安定解の族とその層構造の存在を示した．これは先行研究を大きく前進させるものであり，新しい技法を発見した．さらに副産物として得られたJoseph-Lundgren指数の複数存在という結果もこれまで考えられてこなかった状況である．これは更なる研究を呼び起こす可能性のある結果であり，興味深いことを見つけたと言える．