A Flower-Shape Geometry and Nonlinear Problems on Strip-Like Domains

Abstract

In the present paper, we show how to define suitable subgroups of the orthogonal group $${O}(d-m)$$ O ( d - m ) related to the unbounded part of a strip-like domain $$\omega \times {\mathbb {R}}^{d-m}$$ ω × R d - m with $$d\ge m+2$$ d ≥ m + 2 , in order to get “mutually disjoint” nontrivial subspaces of partially symmetric functions of $$H^1_0(\omega \times {\mathbb {R}}^{d-m})$$ H 0 1 ( ω × 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 $${\mathbb {R}}^d$$ R d , as for instance, the ones due to Bartsch and Willem. The techniques used here are new.

On nonlocal Dirichlet problems with oscillating term

“…Following the seminal work of Ricceri [31], an impressive number of publications appeared, most of them dedicated to the study of suitable extensions of his variational principle as well as of its consequences; see, for instance, the books [16,21,23] and the references therein. Recent applications of [31, Theorem 2.1] can be found in [6,19,22]. See also [20] for related topics.…”