Positive solutions to a supercritical elliptic problem which concentrate along a thin spherical hole

• Published in: arXiv.org
• Main Authors: Clapp, Mónica, Faya, Jorge, Pistoia, Angela
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 06.04.2013
• Subjects: Domains
• ISSN: 2331-8422

We consider the supercritical problem \[ -\Delta v=|v|^{p-2}v in \Theta_{\epsilon}, v=0 on \partial\Theta_{\epsilon}, \] where \(\Theta\) is a bounded smooth domain in \(\mathbb{R}^{N}\), \(N\geq3\), \(p>2^{\ast}:=2N/(N-2)\), and \(\Theta_{\epsilon}\) is obtained by deleting the \(\epsilon\)-neighborhood of some sphere which is embedded in \(\Theta\). In some particular situations we show that, for \(\epsilon>0\) small enough, this problem has a positive solution \(v_{\epsilon}\) and that these solutions concentrate and blow up along the sphere as \(\epsilon\) tends to 0. Our approach is to reduce this problem to a critical problem of the form \[ -\Delta u=Q(x)|u|^{4/(n-2)}u in \Omega_{\epsilon}, u=0 on \partial\Omega_{\epsilon}, \] in a punctured domain \(\Omega_{\epsilon}:=\{x\in\Omega:|x-\xi_{0}|>\epsilon\}\) of lower dimension, by means of some Hopf map. We show that, if \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^{n}\), \(n\geq3\), \(\xi_{0} is in\Omega,\) \(Q is in C^{2}(\b{\Oarmega})\) is positive and \(\nabla Q(\xi_{0})\neq0\) then, for \(\epsilon>0\) small enough, this problem has a positive solution \(u_{\epsilon}\), and that these solutions concentrate and blow up at \(\xi_{0}\) as \(\epsilon\) goes to 0.