Blow-up solutions of some nonlinear elliptic problems
Blow-up solutions of some nonlinear elliptic problems
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DOI：
10.5209/rev_rema.1997.v10.n2.17459
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发表时间：
1997
影响因子：
0.8
通讯作者：
H. Brezis;J. Vázquez
中科院分区：
文献类型：
--
作者：
H. Brezis;J. Vázquez
设f（s）/s - 00为a -，且f（s）/s - 00为连续的、正的、递增的、凸的函数f an [O，oc），且边界光滑，Dirichiel dala u 100 = O. oc>.在这些条件下，I是参数A > O的最大值或极值，使得该问题有解。我们研究了当Ihe>'是无端点的（即，奇异解或爆破解）。我们用两个条件的标准化刻画了奇异H_i极值解和外值：（i）它们必须是L中的极值，（Ii）必须满足一个哈代不等式，该不等式传递了线性化算子的第一特征值是无穷大的这一事实.为了将这一特征化应用于文学中的典型例子，我们需要对经典的哈代不等式进行改进，使之成为常数。我们建立了这样一个结果，它是n > 2的任意方向上的哈代和庞加莱不等式的同时推广.一个引人注目的例子unbunded exiremal solution是这样一个事实，即它们周围的问题的线性化恰好是形式上可逆的，然而逆函数和隐函数的应用却不能产生通常的连续结果。我们认为这是一个问题，并解释为缺乏适当的功能设置这一现象。
posed in a baunded domain fi of R~ with smooth boundary 8 SI wiih Dirichiel dala u100 = O, and a continuous, positive, increasing and convex funetion f an [O,oc) such thai f(s)/s — 00 as a —. oc>. Under Ihese conditions Ihere is a maximal or extremal value of the parameter A > O such thaI ihe problem has a solution. We invesligate Ihe exisience and properties of Ihe corresponding extrema) solutions when Ihe>’ are unhaunded (i.e., singular or blow-up solutions). We characterize ihe singular Hí extremal solulions and ihe extrenial value by a eriterioncansisiing of twa condiiions: (i) they musí be ener~’ saluiions, mal in L~; (Ii) tite>’ musí sntisfy a Hardy inequalil>’ which transíates Ihe fact thai ihe firsí eigenvalue of the linearized aperator is nannegalive. In arder to apply ibis characlerizatiat to ihe typical examples arising in Ihe lilerature we need an improved version of ihe cíassical Hardy inequalil>’ wiih besí constaní. We aiablish such a resulí as a simultaneous generalization of Hardy’s and Poincaré’s inequaUiies for ah dirnensions n > 2. A striking prapert>’ of sorne examples of unbaunded exiremal solutiona is the fact thai tite hinearization of ihe prablem araund them happens to be formalí>’ invertible and nevertheless tite npplication of ihe Inverse and Imphicii Funetion iheorems falís to produce Ihe usual exislence ar continuation resulís. We consider ihis question and explain tite pitenomenon as a lack of appropriate funetional setting.