Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results

IF 1 3区 数学 Q1 MATHEMATICS
Tomás Sanz-Perela
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引用次数: 0

Abstract

We study stable solutions to fractional semilinear equations \((-\Delta )^s u = f(u)\) in \(\Omega \subset {\mathbb {R}}^n\), for convex nonlinearities f, and under the Dirichlet exterior condition \(u=g\) in \({\mathbb {R}}^n {\setminus } \Omega\) with general g. We establish a uniqueness and a classification result, and we show that weak (energy) stable solutions can be approximated by a sequence of bounded (and hence regular) stable solutions to similar problems. As an application of our results, we establish the interior regularity of weak (energy) stable solutions to the problem for the half-Laplacian in dimensions \(1 \leqslant n \leqslant 4\).

分数半线性方程的稳定解:唯一性、分类和近似结果
我们研究了凸非线性 f,在一般 g 的情况下,分式半线性方程 \((-\Delta )^s u = f(u)\) in\(\Omega \子集 {\mathbb {R}}^n\) 的稳定解,以及 Dirichlet 外部条件 \(u=g\) in\({\mathbb {R}}^n {\setminus }\Omega\) 下的稳定解。我们建立了一个唯一性和一个分类结果,并证明弱(能量)稳定解可以由类似问题的有界(因而规则)稳定解序列近似得到。作为我们结果的一个应用,我们建立了维数(1 \leqslant n \leqslant 4\ )中半拉普拉奇问题的弱(能量)稳定解的内部正则性。
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来源期刊
CiteScore
2.10
自引率
10.00%
发文量
99
审稿时长
>12 weeks
期刊介绍: This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it). A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.
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