On existence of solutions for differential equations and their properties via functional analysis

通过泛函分析论微分方程解的存在性及其性质

• 批准号: 13640158
• 负责人: SHIOJI Naoki
• 金额: $ 2.11万
• 依托单位: Yokohama National University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640158/
• 关键词: evolution equation; subtangential; periodic; variational method; subharmonic; singular elliptic problem; multiple solutions; 変分法; 特異楕円型方程式; nonsmooth analysis; Brezis-Nirenberg型定理; 正値解の多重性; 正値解; 解の多重性; subharmonic solutions; variational met

We studied the existence of solutions of both initial value problem and periodic problem for the evolution equation u'(t)+Au(t)∋f(t,u(t)), 0【less than or equal】t【less than or equal】T under the condition u(t)∈K. Here, (V,‖・‖) is a reflexive Banach space which is densely imbedded into a Hilber space (H,<・,・>), A⊂・H×H is a maximal monotone operator satisfying <Ax-Ay,x-y>【greater than or equal】‖x-y‖^p (p>1), K is a closed, convex subset of H, and f : [0,T]×K→H is a Caratheodory mapping. Since the condition for continuity of f(t,・) is on the topology of V, our result is an extension of Bothe's. We also assumed a subtangential condition for f. Moreover, we assumed that the metric projection P : H→K satisfies P(V)⊂V and P : (V,‖・‖) →(V,‖・‖) is continuous in order to associate the continuity of f and the topology of H. We showed that these assumptions are natural in applications.We showed existence of subharmonic solutions for the sysytem of second order ordinary equation u(t)+G'(u(t))=f(t), t … More ∈R in R^N. Here, N is a natural number, f ∈C(R,R^N) is a T-periodic function satisfying (1/T) ∫^T_0 f(t)dt=0,and G∈C^2(R^N,R) a function without convexity. Using Morse's inequality, in the case when the norm of f is sufficiently small, we also showed that there exist at least two solutions which are κT-periodic but not T-periodic for each sufficiently large prime number.Under the homogeneous Dirichlet boundary condition, we studied existence of multiplicity of positive solutions for a singular elliptic equation -Δu=λu^<-q>+u^p in Ω. Here, Ω is a bounded domain in R^N, λ>0,q>0 and p>1. In the case of 0<-q<1,our result is related to Ambrosetti-Brezis-Cerami's. Since behaviors for associated functionals are similar, we can naturally expect that a similar result holds ; so in the case when λ>0 is sufficiently small, we showed that the problem has at least two positive solutions. We also showed that the obtained positive solutions are clasical under some assumption. The assumption is less restrictive the condition that ∂Ω is C^2. Less

研究了演化方程u'(t)+Au(t)∋f(t,u(t))的初值问题和周期问题解的存在性，0【小于等于】t【小于或等于】T，条件为 u(t)∈K 此处，(V,‖・‖) 是一个自反巴纳赫空间，它稠密地嵌入到希尔伯空间 (H,<・,・>) 中， A⊂・H×H 是满足 <Ax-Ay,x-y>【大于或等于】‖x-y‖^p (p>1) 的最大单调算子，K 是 H 的闭凸子集，f : [ 0,T]×K→H 是 Caratheodory 映射，由于 f(t,·) 的连续性条件位于 V 的拓扑上，因此我们的结果是 Bothe 的扩展。此外，为了关联，我们假设度量投影 P : H→K 满足 P(V)⊂V 并且 P : (V,‖·‖) →(V,‖·‖) 是连续的。 f 的连续性和 H 的拓扑。我们证明了这些假设在应用中是自然的。我们证明了二阶常方程组的次调和解的存在性u(t)+G'(u(t))=f(t), t … More ε R in R^N 这里，N 是自然数，f εC(R,R^N) 是 T。 - 满足 (1/T) ∫^T_0 f(t)dt=0 的周期函数，并且 G∈C^2(R^N,R) 是一个没有凸性的函数，在 f 的范数的情况下。是当足够小时，我们还证明对于每个足够大的素数，至少存在两个κT周期但不是T周期的解。在齐次Dirichlet边界条件下，我们研究了奇异椭圆方程正解重数的存在性-Δu=λu^<-q>+u^p in Ω 此处，Ω 是 R^N 中的有界域，在 λ>0、q>0 且 p>1 的情况下。 0<-q<1，我们的结果与 Ambrosetti-Brezis-Cerami 的结果相关，由于相关泛函的行为相似，我们自然可以预期类似的结果成立；因此，当 λ>0 足够小时，我们证明了。我们还表明，在某些假设下，所获得的正解是经典的，即 ∂Ω 为 C^2。