Let ▫$m \ge 1$▫ and ▫$d \ge 2$▫ be integers and consider a strip-like domain ▫$\mathcal{O} \times \mathbb{R}^d$▫, where ▫$\mathcal{O} \subset \mathbb{R}^m$▫ is a bounded Euclidean domain with smooth ...boundary. Furthermore, let ▫$p \colon \hat{\mathcal{O}} \times \mathbb{R}^d \to \mathbb{R}$▫ be a uniformly continuous and cylindrically symmetric function. We prove that the subspace of ▫$W^{1, p(x,y)} (\mathcal{O} \times \mathbb{R}^d)$▫ consisting of the cylindrically symmetric functions is compactly embedded into ▫$L^\infty (\mathcal{O} \times \mathbb{R}^d)$▫ provided that ▫$$m+d < p_{-} := \inf_{(x,y) \in\hat{\mathcal{O}} \times \mathbb{R}^d} p(x,y) \le p_{+} := \sup_{(x,y) \in\hat{\mathcal{O}} \times \mathbb{R}^d} p(x,y) < +\infty.$$▫ As an application, we study a Neumann problem involving the ▫$p(x,y)$▫-Laplacian operator and an oscillating nonlinearity, proving the existence of infinitely many cylindrically symmetric weak solutions. Our approach is based on variational and topological methods in addition to the principle of symmetric criticality.
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