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Existence of clustering high dimensional bump solutions of superlinear elliptic problems on expanding annuli. (English) Zbl 1285.35047

Summary: We consider the nonlinear elliptic problem \[ -{\Delta}u = u^p \quad \text{in } {\Omega}_R, \quad u > 0 \quad \text{in } {\Omega}_R, \quad u = 0 \quad \text{in } {\Omega}_R \] where \(p > 1\) and \({\Omega}_R = \{x \in \mathbb R^N : R < |x| < R + 1\}\) with \(N \geqslant 3\). It is known that, as \(R \to \infty\), the number of nonequivalent solutions of the above problem goes to \(\infty\) when \(p \in (1,(N + 2)/(N - 2))\), \(N \geqslant 3\). Here we prove the same phenomenon for any \(p > 1\) by finding \(O(N - 1)\)-symmetric clustering bump solutions which concentrate near the set \(\{(x_1, \dots, x_N) \in {\Omega}_R : x_N = 0\}\) for large \(R > 0\).

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
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