Compact embedding of a degenerate Sobolev space and existence of entire solutions to a semilinear equation for a Grushin-type operator (Q2568691)

The authors consider the Sobolev space \[ S^{1,2}(\mathbb{R}^N) = \{u \in L^2(\mathbb{R}^N) : \nabla_Gu \in L^2(\mathbb{R}^N)\}, \] where div\((\nabla_G)\) is the Grushin operator \(\Delta_G = \Delta_x + |x|^{2\alpha}\Delta_y\), with \(\alpha > 0\), \(N = n+m\), and \((x,y) \in\mathbb{R}^n \times\mathbb{R}^m\). It is shown that the embedding \[ C \to L^q(\mathbb{R}^N) \] is compact for \(2 < q <2^*_G\), where \(C\) is the cone \[ C = \{u \in S^{1,2}(\mathbb{R}^N) : u(x,y) = \phi(|x|,|y|) \text{ for some \(\phi\) non-increasing in } y\}. \] This result is then applied to establish the existence of a positive solution, radially symmetric with respect to \(x\) and \(y\), of the problem \[ -\Delta_Gu + \lambda u = u^{q-1} \] for \(\lambda > 0\) and \(q \in (2,2^*_G)\).