{"title":"On the Nodal Set of Solutions to Some Sublinear Equations Without Homogeneity","authors":"Nicola Soave, Giorgio Tortone","doi":"10.1007/s00205-024-01970-4","DOIUrl":null,"url":null,"abstract":"<div><p>We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem </p><div><div><span>$$\\begin{aligned} -\\Delta u = \\lambda _+(u^+)^{p-1}-\\lambda _-(u^-)^{q-1}, \\end{aligned}$$</span></div></div><p>where <span>\\(1 \\le p<q<2\\)</span>, <span>\\(\\lambda _+ >0\\)</span>, <span>\\(\\lambda _- \\ge 0\\)</span>. The equation is characterized by the sublinear <i>inhomogeneous</i> character of the right hand-side, which makes it difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 2","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-024-01970-4","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem
$$\begin{aligned} -\Delta u = \lambda _+(u^+)^{p-1}-\lambda _-(u^-)^{q-1}, \end{aligned}$$
where \(1 \le p<q<2\), \(\lambda _+ >0\), \(\lambda _- \ge 0\). The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes it difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.
Abstract We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem $$\begin{aligned} -\Delta u = \lambda _+(u^+)^{p-1}-\lambda _-(u^-)^{q-1}, \end{aligned}$$ 其中 \(1 \le p<;q<2\) ,\(\lambda _+ >0\) ,\(\lambda _-\ge 0\) 。该方程的特点是右边具有亚线性非均质性,这使得它很难以标准方式适应自由边界问题的经典工具,如单调性公式和炸毁论证。我们的主要结果是:接近结点集的解的局部行为;可接受消失阶的完整分类,以及对局部最小值奇异集豪斯多夫维度的估计;退化(非局部最小)解的存在。
期刊介绍:
The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.