In the present paper, we show how to define suitable subgroups of the orthogonal group \({O}(d-m)\) related to the unbounded part of a strip-like domain \(\omega \times {\mathbb {R}}^{d-m}\) with \(d\ge m+2\), in order to get “mutually disjoint” nontrivial subspaces of partially symmetric functions of \(H^1_0(\omega \times {\mathbb {R}}^{d-m})\) which are compactly embedded in the associated Lebesgue spaces. As an application of the introduced geometrical structure, we prove (existence and) multiplicity results for semilinear elliptic problems set in a strip-like domain, in the presence of a nonlinearity which either satisfies the classical Ambrosetti–Rabinowitz condition or has a sublinear growth at infinity. The main theorems of this paper may be seen as an extension of existence and multiplicity results, already appeared in the literature, for nonlinear problems set in the entire space \({\mathbb {R}}^d\), as for instance, the ones due to Bartsch and Willem. The techniques used here are new.

在本文中，我们展示了如何定义与带状域\（\ omega \ times {\ mathbb {R}}的无界部分相关的正交组\（{O}（dm）\）的合适子组^ {dm} \）与\（d \ ge m + 2 \），以获取\（H ^ 1_0（\ omega \ times {\ mathbb {R}}的部分对称函数的“互不相交”非平凡子空间^ {dm}）\）它们紧密地嵌入相关的Lebesgue空间中。作为引入的几何结构的一种应用，我们证明了存在线性的非线性存在下（存在或存在）多重结果，该半线性椭圆问题设置在带状域中，该非线性要么满足经典的Ambrosetti–Rabinowitz条件，要么在无限。对于整个空间\（{\ mathbb {R}} ^ d \）中设置的非线性问题，本文的主要定理可以看作是存在和多重结果的扩展，已经存在于文献中，例如，那些是由于Bartsch和Willem。这里使用的技术是新的。