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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 $$\partial \Omega$$.

##### MSC:
 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