Abstract
We consider the elliptic problem Δu + u p = 0, u > 0 in an exterior domain, \({\Omega = \mathbb{R}^N{\setminus}\mathcal{D}}\) under zero Dirichlet and vanishing conditions, where \({\mathcal{D}}\) is smooth and bounded in \({\mathbb{R}^N}\) , N ≥ 3, and p is supercritical, namely \({p > \frac{N+2}{N-2}}\) . We prove that this problem has infinitely many solutions with slow decay \({O(|x|^{-\frac2{p-1}})}\) at infinity. In addition, a solution with fast decay O(|x|2-N) exists if p is close enough from above to the critical exponent.
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Dávila, J., del Pino, M., Musso, M. et al. Fast and slow decay solutions for supercritical elliptic problems in exterior domains. Calc. Var. 32, 453–480 (2008). https://doi.org/10.1007/s00526-007-0154-1
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DOI: https://doi.org/10.1007/s00526-007-0154-1