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Positive solutions to a supercritical elliptic problem that concentrate along a thin spherical hole. (English) Zbl 1319.35016

Summary: We consider the supercritical problem \[ - \Delta v = \left| v \right|^{p - 2}v \quad \text{ in } \Theta_\epsilon, \quad v = 0 \text{ on }\partial \Theta_\epsilon, \] where \(\Theta\) is a bounded smooth domain in \(\mathbb{R}^N\), \(N \geq 3\), \(p > 2^\ast := 2N/(N - 2)\), and \(\Theta_\epsilon\) is obtained by deleting the \(\epsilon\)-neighborhood of some sphere which is embedded in \(\Theta\). We show that in some particular situations, for small enough \(\epsilon > 0\), this problem has a positive solution \(v_\epsilon\) and that this solution concentrates and blows up along the sphere as \(\epsilon \to 0\). Our approach is to reduce this problem by means of a Hopf map to a critical problem of the form \[ - \Delta u = Q(x) \left| u \right|^{4/n - 2}u \quad \text{ in } \Omega_\epsilon, \quad u = 0 \text{ on }\partial \Omega_\epsilon, \] in a punctured domain \(\Omega_\epsilon : = \{ x \in \Omega :\left| x - \xi_0 \right| > \epsilon \} \) of lower dimension. We show that if \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^n\), \(n \geq 3\), \(\xi_0 \in \Omega\), \(Q \in C^2(\overline \Omega)\) is positive, and \(\nabla Q(\xi_0) \neq 0\), then for small enough \(\epsilon > 0\), this problem has a positive solution \(u_\epsilon\) which concentrates and blows up at \(\xi_0\) as \(\epsilon \to 0\).

MSC:

35J20 Variational methods for second-order elliptic equations
35B09 Positive solutions to PDEs
35B44 Blow-up in context of PDEs
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