Concentration on submanifolds for an Ambrosetti–Prodi type problem

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• 作者：Zied Khemiri ; Fethi Mahmoudi…
• 关键词：Mathematics Subject Classification35J25 ; 35J20 ; 35B33 ; 35B40
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：56
• 期：2
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0835
• 卷排序：56

文摘

Given a smooth bounded domain \(\Omega \) of \(\mathbb {R}^n\) and consider the problem $$\begin{aligned} \left\{ \begin{array} {ll} - \Delta u = |u|^p - t\,\psi &{}\quad \hbox { in } \ \Omega \\ u = 0 &{}\quad \hbox { on }\ \partial \Omega \end{array}\right. \end{aligned}$$ (0.1)where t is a large positive parameter, \(p>1\) and \(\psi \) is an eigenfunction of \(- \Delta \) with Dirichlet boundary condition corresponding to the first eigenvalue \(\lambda _1\). Assuming that \(\Omega \) contains a k-dimensional compact submanifold K which is stationary and non-degenerate for the weighted functional $$\begin{aligned} \int _K \psi ^{(1 - \frac{1}{p})(\frac{p+1}{p-1}-\frac{n-k}{2})}dvol \end{aligned}$$such that \(\mathrm{dist}(K,\partial \Omega )>\delta _0>0\), then for \(1< p < \frac{n+2-k}{n-2-k}\) we prove the existence of a sequence \(t=t_j\rightarrow \infty \) and solutions \(u_t\) that concentrate along K. This result proves in particular the validity of a conjecture by Hollman–Mckenna in full generality, see Hollman and McKenna (Commun Pure Appl Anal 10(2):785–802, 2011), extending the result by Manna and Santra (Discrete Contin Dyn Syst 36(10):5595–5626, 2016) where the case \(n=2\) and \(k=1\) has been considered.Mathematics Subject Classification35J2535J2035B3335B40Communicated by A. Malchiodi.