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The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents. (English) Zbl 0829.35041
Let \(\Omega \subset \mathbb{R}^N\) \((N \geq 3)\) be a bounded domain with smooth boundary and let \(p = (N + 2)/(N - 2)\). The author is concerned with the existence and multiplicity of solutions \(u\) to the Neumann problem \[ - \Delta u + \lambda u = |u |^{p - 1} u \text{ in } \Omega, \quad \partial u/ \partial \nu = 0 \text{ on } \partial \Omega, (I)_\lambda \] where \(\lambda\) is a positive constant and \(\nu\) is the unit outer normal to \(\partial \Omega\). Denote also by \(H(x)\) the mean curvature of \(\partial \Omega\) at \(x\) with respect to \(\nu\). The aim of the paper is to establish connections between the topology of \(\partial \Omega\) and the number of nonconstant positive solutions \(u\) to \((I)_\lambda\). For instance, the author shows that, if \(H(x) > 0\) on \(\partial \Omega\) and \(\lambda\) is sufficiently large, then problem \((I)_\lambda\) possesses at least \(\text{cat} (\partial \Omega)\) nonconstant positive solutions. Actually, this is a corollary of a more general result, still involving the topology of \(\partial \Omega\).

35J65 Nonlinear boundary value problems for linear elliptic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
37G99 Local and nonlocal bifurcation theory for dynamical systems