×

Low-dimensional compact embeddings of symmetric Sobolev spaces with applications. (English) Zbl 1223.46036

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.

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)
PDFBibTeX XMLCite
Full Text: DOI