Sobolev embeddings involving symmetry (Q855489)

This paper deals with domains \(\Omega=\Omega_1\times \Omega_2\), where \( \Omega_1\subset \mathbb R^m\), \(m\geq 1\), is a bounded regular domain, and \(\Omega_2\) is a \(k\geq 2\)-dimensional ball of radius \(R\), centered at the origin. Let \(\overset{o}{H}^1_s(\Omega)= \{u\in H^1_0(\Omega):u(.,x_2)=u(.,| x_2| ),\forall x_2\in \Omega_2\}\). If \(h\geq 0\) is a Hölder continuous function on \(\overline{\Omega},\) \(L^p_h(\Omega)\) denotes the weighted \(L^p\) space equipped with the norm \[ \|u\|_{h,p}= \biggl(\int_{\Omega}h| u| ^p\biggr)^{\frac{1}{p}}. \] It is proved that, for \(h(x)=| x_2| ^l\), where \(l\) is a positive real number, the embedding \(\overset{o}{H}^1_s(\Omega{)} L^q_h(\Omega)\) is compact for \(q\in (1,\frac{2n}{n-2}+\tau)\) for a suitable \(\tau>0\) which depends on \(m, k\) and \(l\), where \( n\) is the dimension of \(\Omega.\) It is mentioned that the result still holds for some more general functions \(h\), and an application to a simple nonlinear elliptic boundary value problem is given.