2017 Fiscal Year Final Research Report
Study on the bifurcation structure of positive solutions for concave-convex mixed nonlinear elliptic boundary value problems with indefinite weights
| Project/Area Number |
15K04945
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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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| Research Field |
Mathematical analysis
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| Research Institution | Ibaraki University |
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Principal Investigator |
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| Research Collaborator |
RAMOS QUOIRIN Humberto Universidad de Santiago de Chile
KAUFMANN Uriel Universidad Nacional de Córdoba
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| Project Period (FY) |
2015-04-01 – 2018-03-31
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| Keywords | 非線形楕円型境界値問題 / concave-convex型非線形性 / 非自明非負解 / 符号不定係数 / 分岐解析 / 正値性 / 優解劣解 / 変分的手法 |
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| Outline of Final Research Achievements |
We study concave-convex nonlinear elliptic boundary value problems, equipped with indefinite weights, in a smooth bounded domain of the Euclidean space, and investigate the existence of nontrivial nonnegative solutions and their properties.
On one hand, we have determined the bifurcation structure of the nontrivial nonnegative solutions set in some cases, as a parameter included varies. Especially, we have obtained a loop type component of nontrivial nonnegative solutions which bifurcates from the trivial solutions line.
On the other hand, we have provided certain sufficient conditions for the positivity of nontrivial nonnegative solutions. The strong maximum principle does work for nonlinear elliptic problems which are regular around zero solutions in the standard sense, in which class any nontrivial nonnegative solution so implies a positive solution. However, it does not work in general for concave-convex problems with indefinite weights.
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| Free Research Field |
非線形偏微分方程式
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