{"title":"Stable solutions to fractional semilinear equations: uniqueness, classification, and approximation results","authors":"Tomás Sanz-Perela","doi":"10.1007/s10231-024-01497-1","DOIUrl":null,"url":null,"abstract":"<p>We study stable solutions to fractional semilinear equations <span>\\((-\\Delta )^s u = f(u)\\)</span> in <span>\\(\\Omega \\subset {\\mathbb {R}}^n\\)</span>, for convex nonlinearities <i>f</i>, and under the Dirichlet exterior condition <span>\\(u=g\\)</span> in <span>\\({\\mathbb {R}}^n {\\setminus } \\Omega\\)</span> with general <i>g</i>. 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 <span>\\(1 \\leqslant n \\leqslant 4\\)</span>.</p>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali di Matematica Pura ed Applicata","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10231-024-01497-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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 \(1 \leqslant n \leqslant 4\).
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.