2023 Fiscal Year Final Research Report
Positive solution sets for nonlinear elliptic boundary value problems of concave-convex mixed type with some singularity
| Project/Area Number |
18K03353
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | Ibaraki University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | 非線形楕円型境界値問題 / sublinear非線形性 / concave-convex非線形性 / 非自明非負解 / 変分解析 / 分岐解析 / 優解劣解の構成 |
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| Outline of Final Research Achievements |
We consider nonlinear elliptic boundary value problems of sublinear nonlinearity with some singularity and evaluate the existence and behavior of the nontrivial nonnegative solutions under the Dirichlet boundary condition as a parameter varies. Then, we consider a class of concave-convex mixed nonlinear problems and prove the existence, nonexistence, and multiplicity of the nontrivial nonnegative solutions. Further, we construct a subcontinuum, i.e., a nonempty, closed, and connected subset, of loop type in the nonnegative solution set. Then, we study the logistic elliptic problem under a sublinear boundary condition with some singularity and clarify the existence, uniqueness, multiplicity, and asymptotic behavior of the nontrivial nonnegative solutions as a parameter varies.
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| Free Research Field |
非線性偏微分方程式
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| Academic Significance and Societal Importance of the Research Achievements |
(1)滑らかな境界をもつ多次元有界領域において，位相的手法を用いてloopをなす解集合の存在を導けた．加えて，線形化固有値問題の第2固有値の解析により，正値解の多重性を伴う可能な限り最小の解曲線を獲得できた．(2)べき乗型非線形項に現れるべき値をパラメータとした非自明非負解の解析は従来の研究ではあまり見られない．べき値がsublinear非線形項の特異性の強さを表現しており，それに応じて非自明非負解の正値性の強さを顕在化できた．(3)sublinear非線形境界条件を用いた沿岸漁業収穫モデルの解析は多次元有界領域における初の試みであった．このモデルへの解析的アプローチの進展が今後期待できる．
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