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Title :符号不定係数と混合型非線形性を伴う楕円型境界値問題の正値解の分岐構造に関する研究
Title alternative :Study on the bifurcation structure of positive solutions for concave-convexmixed nonlinear elliptic boundary value problems with indefinite weights
Authors :梅津, 健一郎
Authors alternative :UMEZU, Kenichiro
Issue Date :14-Jun-2018
Abstract :研究成果の概要(和文):多次元ユークリッド空間の滑らかな有界領域において,concave-convex 混合型の非線形性と符号不定係数を兼ね備える非線形楕円型境界値問題に対して非自明非負解の存在その性質を研究した.研究は2つの方向で進められ,それぞれ次の成果を得た.(1) パラメータの変化にしたがった非自明解集合の大域的位相構造を決定した.特に,ある条件のもとでループ形状の連続体位相構造の存在を示した.(2) concave 性と符号不定係数の組み合わせにより発生する非自明解の正値性問題に対して,肯定的に解決するための十分条件を得た(零解のまわりで十分な滑らかさをもつ非線形問題に対しては非自明解は正値性をもつ). 研究成果の概要(英文):We study concave-convex nonlinear elliptic boundary value problems, equippedwith 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.
Type Local :科研等報告書
Publisher :茨城大学
URI :http://hdl.handle.net/10109/13966
Appears in Collections:Grants-in-aid for scientific research Report (College of Education)

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