Compact imbedding theorems with symmetry of Strauss-Lions type for the space \(W^{1,p(x)}(\Omega)\) (Q5929383)

For an open domain \(\Omega\) of \(\mathbb{R}^N\) denote: \(L^\infty_+(\Omega)=\{p\in L^\infty(\Omega): \inf[p(x):x\in \Omega]<1\}\). If \(p\in L^\infty_+(\Omega)\), then the set \(L^{p(x)} (\Omega)=\{u: u\) is real, measurable \(\int_\Omega|u(x)|^{p(x)}dx<\infty\}\) is a Banach space equipped with the norm \(|u|_{p(x)} = \inf\{\lambda>0:\int_\Omega|u/\lambda|^{p(x)}dx\leq 1\}\) (in fact, it is a special kind of generalized Orlicz space). Moreover, we define the generalized Lebesgue-Sobolev space \(W^{1,p(x)}(\Omega)\) as the set \(W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega):|\nabla u|\in L^{p(x)}(\Omega)\}\) equipped with the norm \(\|u\|=|u|_{p(x)} + |\nabla u|_{p(x)}\).NEWLINENEWLINENEWLINEIn the paper a result on the compact imbedding of the space \(W^{1,p(x)}(\Omega)\) into \(L^{\alpha(x)}(\mathbb{R}^N)\) is proved, under the assumptions that \(p: \mathbb{R}^N\to\mathbb{R}\) is uniformly continuous, radially symmetric \(\alpha:\mathbb{R}^N\to\mathbb{R}\) is measurable and satisfies some extra requirements.