A flower-shape geometry and nonlinear problems on strip-like domains

In the present paper we show how to define suitable subgroups of the orthogonal group related the unbounded part of a strip-like domain, in order to get ``mutually disjoint'' nontrivial subspaces of partially symmetric functions of $H^1_0(omega imes RR^{d-m})$ 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 presence of a nonlinearity which either satisfies the classical Ambrosetti-Rabinowitz condition or has a sublinear growth at infinity. The main theorems got along 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, as for instance, the ones obtained by Bartsch and Willem. The techniques used are new.