## 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

### Keywords:

supercritical elliptic problem; positive solution
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### References:

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