The Sobolev inequality for Paneitz operator on three manifolds
Let \((M, g)\) be a smooth compact Riemannian manifold of dimension \(3\). The Paneitz operator is defined by \[ P \phi = \Delta^2 \phi + 4 \operatorname {div} (Rc(\nabla \phi, e_i)e_i) - \frac{5}{4} \operatorname {div} (R \nabla \phi) - \frac{1}{2} Q \phi,\quad \forall \phi \in C^{\infty}(M, \mathbb R), \] where \(Rc\) is the Ricci operator, \(\{e_1, e_2, e_3\}\) is an arbitrary local orthonormal frame, and \(Q\) is the curvature, \[ Q=-\frac{1}{4} \Delta R - 2 | Rc | ^2 + \frac{23}{32} R^2. \] One notes that \(P\) satisfies the conformal covariant property, \[ P_{{\rho}^{-4}g} \phi = {\rho}^7 P_g (\rho \phi),\quad \forall \rho \in C^{\infty} (M, \mathbb R),\;\rho >0,\quad \forall \phi \in C^{\infty} (M, \mathbb R). \] In particular, if \(\widetilde{g} = u^{-4}g\) for some \(u \in C^\infty (M), u >0\), one has \(\widetilde{Q}=-2u^7P_g u\), hence the problem of finding a conformal metric with constant \(Q\)-curvature becomes \[ P_gu={\text{constant}}\cdot u^{-7},\quad {\text{on}}\;M,\quad {\text{and}} \quad u \in C^{\infty} (M),\quad u>0. \] This problem has also a variational definition. Let \[ E(u,v)=\int_M \left( \Delta u \Delta v - 4Rc(\nabla u, \nabla v) + \frac{5}{4} R \nabla u \cdot \nabla v - \frac{1}{2} Q u v \right)\, d \mu, \quad \forall u, v \in H^2(M), \] and \[ I(u)=\frac{E(u,u)}{| u| ^2_{L^{-6}(M)}}= E(u,u) | u^{-1}| ^2_{L^6(M)},\quad \forall u \in H^2(M),\;u>0. \] Thus, the previous description is the Euler-Lagrange equation for \(I\). To find a critical point, it suffices to minimize the functional \(I\) over the class of positive functions in \(H^2(M)\). Its infimum, although a conformally invariant quantity, is no longer easily seen to satisfy \(Y_4^3(M, g):=\inf_{u \in H^2(M), u>0} I(u)> -\infty.\) This was trivial in the case of the analogous Yamabe constant where one simply applies the usual Sobolev inequality. The negative exponent of the present problem does not allow the same reasoning here. To deal with this, the following condition is introduced: We say that \((M, g)\) satisfies the condition \((P+)\) if \(\forall u \in H^2(M)\setminus \{0 \},\;u \geq 0\) and \(u\) vanishes somewhere, we have \(E(u,u) >0\). (Note that the condition \((P+)\) is satisfied when \(P_g>0\), a situation studied earlier by \textit{X. Xu} and \textit{P. C. Yang} [Discrete Contin. Dyn. Syst. 7, No. 2, 329--342 (2001; Zbl 1032.58018)], concluding also that, in this case, the critical function is unique, up to a positive constant factor.) The paper's main theorem states that the condition \((P+)\) ensures the existence of the minimizer to \(I(u)\) and guarantees that it is \(> - \infty\). Moreover, the authors show first how certain pointwise curvature conditions imply the positivity of the Paneitz operator. They study the existence and uniqueness of minimizers of \(I(u)\) when \(P_g \geq 0\). In particular, it is shown that for Berger spheres, the minimizing problem always has a solution.
scientific article; zbMATH DE number 2113925