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Multibump solutions for an elliptic problem in expanding domains. (English) Zbl 1011.35059

Let \(\Omega\) be a bounded convex domain in \(\mathbb{R}^N\) with \(C^2\) boundary \(\partial\Omega\) and \(\nu(x)\) be the outward unit normal of \(\partial \Omega\) at \(x\). Define \[ \Omega_R= \bigl\{x\mid x=\widetilde x+t\nu(\widetilde x),\;\forall\widetilde x\in\partial\Omega,\;t\in[R,R+1] \bigr\}. \] The aim of this paper is to construct multibump solutions for an elliptic problem on this expanding domain, such that all the local maximum points of the solution are close to the set \(\{x\mid x=\widetilde x+(R+1/2) \nu(\widetilde x)\), \(\widetilde x\in \partial\Omega\}\) if \(R>0\) is large enough.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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