Exact Morse index computation for nodal radial solutions of Lane–Emden problems

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• 作者：Francesca De Marchis ; Isabella Ianni ; Filomena Pacella
• 关键词：Mathematics Subject Classification35B05 ; 35B06 ; 35J91
• 刊名：Mathematische Annalen
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：367
• 期：1-2
• 页码：185-227
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1807
• 卷排序：367

文摘

We consider the semilinear Lane–Emden problem where B is the unit ball of \(\mathbb {R}^N\), \(N\ge 2\), centered at the origin and \(1<p<p_S\), with \(p_S=+\infty \) if \(N=2\) and \(p_S=\frac{N+2}{N-2}\) if \(N\ge 3\). Our main result is to prove that in dimension \(N=2\) the Morse index of the least energy sign-changing radial solution \(u_p\) of \(({\mathscr {E}}_{p})\) is exactly 12 if p is sufficiently large. As an intermediate step we compute explicitly the first eigenvalue of a limit weighted problem in \(\mathbb {R}^N\) in any dimension \(N\ge 2\).Mathematics Subject Classification35B0535B0635J91Research partially supported by: University of Sapienza funds “Avvio alla ricerca 2015”, PRIN 201274FYK7\(\_005\) grant and INDAM-GNAMPA.