Sobolev inequalities in 2-D hyperbolic space: A borderline case
Let \(\mathbb R^2_+ = \{(x,y) \in \mathbb R^2 : y \in (0,\infty)\}\), \(q\in (2,\infty)\) and \(R\in (-\infty, 1/4)\). Put \[ C(2,q,R) = \inf_u V(u,R,q), \tag{1} \] where \[ V(u,R,q) = \frac{\int_{\mathbb R^2_+} (u^2_x + u^2_y) dx dy - R \int_{\mathbb R^2_+} u^2 y^{-2} dx dy}{(\int_{\mathbb R^2_+} |u|^q y^{-2} dx dy)^{2/q}}\tag{2} \] and the infimum is taken over a set of convenient test functions \(u\not\equiv 0\). The main results of the paper state that there exists \(u\), \(u\not\equiv 0\), such that \[ C(2,q,R) = V(u,R,q) \tag{3} \] and that \(u\) satisfies the corresponding Euler equation, \(\Delta u + Ru + |u|^{q-2} u = 0\), implied by (1) - (3) and a suitable normalization. Moreover, \(C(2,3,0) = (2\pi)^{1/3}\) and \(C(2,4,0) = (8\pi/3)^{1/2}\). It is also shown that \(C(2,q,R)\) coincides with the largest constant \(C\) in the inequality \[ ((-\Delta - R) u, u)_{L^2(\mathbb H^2)} \geq C \|u\|^2_{L^q (\mathbb H^2)} \] for every test function \(u\), where \(\mathbb H^2\) is \(\mathbb R^2_+\) equipped with the Poincaré-Bergman metric \[ y^{-2} [(dx)^2 + (dy)^2]. \]
scientific article; zbMATH DE number 1309141