Asmptotic behaviors of nodal radial solutions to semilinear elliptic equations with Trudinger-Moser critical growth

2023 Fiscal Year Final Research Report

• Project/Area Number: 17K14214
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Muroran Institute of Technology
• Principal Investigator: Naimen Daisuke 室蘭工業大学, 大学院工学研究科, 准教授 (20783278)
• Project Period (FY): 2017-04-01 – 2024-03-31
• Keywords: 解の集中現象

Outline of Final Research Achievements

We accomplished the analysis of asymptotic behaviors of the least energy nodal radial solutions to semilinear elliptic equations with Trudinger-Moser critical growth. We found that the positive part concentrates at the maximum point while the negative part strongly converges to a positive solution. Interestingly, this result implies that a sign-changing solution consists of the noncompact and compact parts. This observation could not be found in the behaviors of positive solutions in the previous works. We also succeeded in the complete classification of the asymptotic behaviors of general nodal radial solutions. In particular, we observed that, interestingly, the solutions may admit multiple concentration parts with multiple compact parts.

Free Research Field

非線型解析

Academic Significance and Societal Importance of the Research Achievements

本研究課題の遂行以前，当該研究分野では当該方程式の「正値」解の爆発挙動が専ら研究されていた。これに対し，本課題は本研究分野において先駆けて「符号変化」爆発解の挙動を本格的に研究したものである。本研究の結果，非コンパクト性を有する臨界方程式特有の集中現象と，それに対照的なコンパクト現象が混在する符号変化爆発解ならではの挙動を捉えることができた。この成果が当該研究分野に与えるインパクトは大きく，今後の符号変化爆発解の研究の発展の起点と成るものといえる。