In this paper, by making use of a new variational principle, we prove existence of nontrivial solutions for two different types of semilinear problems with Neumann boundary conditions in unbounded domains. Namely, we study elliptic equations and Hamiltonian systems on the unbounded domain ##IMG## [http://ej.iop.org/images/0951-7715/33/9/4568/nonab8bacieqn1.gif] {${\Omega}={\mathbb{R}}^{m}{\times}{B}_{r}$} where B r is a ball centred at the origin with radius r in ##IMG## [http://ej.iop.org/images/0951-7715/33/9/4568/nonab8bacieqn2.gif] {${\mathbb{R}}^{n}$} . Our proofs consist of several new and novel ideas that can be used in broader contexts.

中文翻译：

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无界域上的超临界Neumann问题

在本文中，通过使用新的变分原理，我们证明了无界域中具有Neumann边界条件的两种不同类型半线性问题非平凡解的存在。即，我们研究无界域上的椭圆方程和哈密顿系统## IMG ## [http://ej.iop.org/images/0951-7715/33/9/4568/nonab8bacieqn1.gif] {$ {\ Omega } = {\ mathbb {R}} ^ {m} {\ times} {B} _ {r} $}，其中B r是在## IMG ## [http：// ej.iop.org/images/0951-7715/33/9/4568/nonab8bacieqn2.gif] {$ {\ mathbb {R}} ^ {n} $}。我们的证明包括可以在更广泛的上下文中使用的几个新颖的想法。