並列タイトル等サマザマナ コウカ オ モツ ヒセンケイ ダエンガタ ホウテイシキ ノ カイコウゾウ ノ ケンキュウ
Samazamana kōka o motsu hisenkei daengata hōteishiki no kaikōzō no kenkyū
Study of the strucure of solutions to nonlinear elliptic equations with various effects
一般注記出版タイプ： VoR
type:text
本研究課題では特異性や非局所効果のある非線形楕円型方程式の解構造の解明を目標とし，研究を行った．得られた成果は，Born-Infeld方程式(特異性のある方程式)の最小化元の正則性と弱解との関係，分数冪ラプラシアンとHardy-Henon型非線形項を持つ方程式(非局所性のある方程式)の安定解の存在や非存在，また安定解の族が層構造を持つことも示すことができた．これら以外にも劣線形項を持つ方程式，1次元Pucci作用素を含む方程式，大きなパラメータを含む楕円型方程式，L2制約条件付き問題それぞれについて解の存在や形状等を調べ明らかにした．
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.
研究種目 : 基盤研究 (C) (一般)
研究期間 : 2019～2023
課題番号 : 19K03590
研究分野 : 偏微分方程式
連携機関・データベース国立情報学研究所 : 学術機関リポジトリデータベース（IRDB）（機関リポジトリ）