A Flower-Shape Geometry and Nonlinear Problems on strip-like domains

In the present paper, first of all we provide a geometrical structure on Sobolev spaces, using the orthogonal group in Rd-m, d;m, 2 in N with d > m, and its natural action on the space we are interested in. Later, we consider a semilinear elliptic problem set in a striplike domain, in presence of a nonlinearity which either satisfyes the classical Ambrosetti-Rabinowitz condition or has a sublinear growth at infinity. In both cases, thanks to classical critical points theory, we prove the existence of a non-trivial weak solution for this equation. Moreover, as an application of the abstract geometric construction, we obtain a multiplicity result for this nonlinear problem. 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 Rd.