Faraci, Francesca; Iannizzotto, Antonio; Kristály, Alexandru Low-dimensional compact embeddings of symmetric Sobolev spaces with applications. (English) Zbl 1223.46036 Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 2, 383-395 (2011). Let \(m\geq 1\), \(n\geq 2\) be integers, \(\mathcal O\subset \mathbb R^m\) be a bounded domain with Lipschitz boundary \(\partial \mathcal O\), \(\Omega=\mathcal O\times\mathbb R^n\), and \(p>m+n\) be a real number. Note that \(\partial\Omega=\partial \mathcal O\times \mathbb R^N\). A function \( u:\mathbb R^N\to \mathbb R\) is radially symmetric if \(u(z_1)=u(z_2)\), for all \(z_1,z_2\in\mathbb R^N\) satisfying \(|z_1|=|z_2|\). Let \(W^{1,p}_c(\Omega)\) [resp. \(W^{1,p}_s(\mathbb R^m\times \mathbb R^n)]\) be the closed subspace of \(W^{1,p}(\Omega)\) [resp. \(W^{1,p}(\mathbb R^m\times \mathbb R^n)\)] of the cylindrically symmetric functions (resp. partially symmetric functions), i.e.,\[ W^{1,p}_c(\Omega)=\{u\in W^{1,p}(\Omega):u(x,\cdot) \text{ is radially symmetric for all } x\in\mathcal O\} \] [resp. \[ \begin{split} W^{1,p}_s(\mathbb R^m\times\mathbb R^n)=\{u\in W^{1,p}(\mathbb R^m\times\mathbb R^n):\\u(x,\cdot),u(\cdot,y) \text{ is radially symmetric for all } x\in\mathbb R^m,y\in\mathbb R^n\}].\end{split} \]The compact embeddings of \(W^{1,p}_c(\Omega)\) [resp. \( W^{1,p}_s(\mathbb R^m\times\mathbb R^n)]\) into \(L^{\infty}(\Omega)\) [resp. \(L^{\infty}(\mathbb R^m\times\mathbb R^n)]\) are proved.These results complete previous related results by P. L. Lions [J. Funct. Analysis, 49, 315–334 (1982 Zbl 0501.46032)] and A. Kristaly and C. Varga [Math. Nachr. 279, 1756–1765 (2005; Zbl 1161.35456)]. Applications to the existence of infinitely many weak solutions to Neumann problems, involving the \(p\)-Laplacian operator, are studied. Reviewer: Denise Huet (Nancy) Cited in 5 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35J92 Quasilinear elliptic equations with \(p\)-Laplacian 47J20 Variational and other types of inequalities involving nonlinear operators (general) Keywords:symmetric Sobolev spaces; compact embeddings; Neumann problem; \(p\)-Laplacian Citations:Zbl 0501.46032; Zbl 1161.35456 PDFBibTeX XMLCite \textit{F. Faraci} et al., Proc. R. Soc. Edinb., Sect. A, Math. 141, No. 2, 383--395 (2011; Zbl 1223.46036) Full Text: DOI