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The role of the boundary in some semilinear Neumann problems. (English) Zbl 0814.35037
The authors study the problem \[ -\Delta u + \lambda^ 2u = u^{p-1} \quad \text{in} \quad \Omega, \quad u > 0 \quad \text{in} \quad \Omega, \quad {\partial u \over \partial \nu} = 0 \quad \text{on} \quad \partial \Omega, \] where \(\Omega\) is a smooth bounded domain in \(R^ N\), \(N \geq 3\), and \(p \in (2,2N/(N-2))\). They show that this problem has at least \((cat\;\partial \Omega) + 1\) solutions if \(\lambda\) is sufficiently large. The key idea is to show, by a concentration analysis argument, that low energy sublevels of an associated functional inherit the topology of \(\Omega\) if \(\lambda\) is large.
Reviewer: J.Urbas (Canberra)

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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