A compact embedding result for anisotropic Sobolev spaces associated to a strip-like domain and some applications

Let m >= 1 and d >= 2 be integers and consider a strip-like domain O x R-d, where O subset of R-m is a bounded Euclidean domain with smooth boundary. Furthermore, let p : (O) over barx R-d -> R be uniformly continuous and cylindrically symmetric function. We prove that the subspace of W-1,W-p(x,W-y)(O x R-d) consisting of the cylindrically symmetric functions is compactly embedded into L-infinity(O x R-d) provided that m + d < p- := inf ((x,y)is an element of<(O)over bar>xRd) p(x, y) <= p(+) := sup ((x,y)is an element of(O) over bar xRd) p(x, y) < +infinity. 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. (C) 2019 Elsevier Inc. All rights reserved.