Existence results for the higher-order \(Q\)-curvature equation (Q6564425)

The authors prove existence results for solutions of the \(Q\)-curvature equation on closed Riemannian manifolds of dimension \(n\geq 2k+1\). The results are primarily obtained under the assumption that the Yamabe invariant of order \(2k\), the Green's function of the corresponding GJMS operator, and the mass of the operator are positive. \N\NFor a Riemannian manifold \((M, g)\) of dimension \(n\) and an integer \(k\), with \(2{k}<{n}\), the GJMS operators are conformally invariant differential operators \(P_{2k}\) of order \(2k\), analogous to the Laplace-Beltrami operator for \(n=2\), and the conformal Laplacian for \(n\geq 3\). \N\NDenote by \(C^{\infty}(M)\) the space of smooth functions on \(M\), and by \(C^{\infty}_+(M)\) the subset of positive ones. The GJMS operator \(P_{2k}\) corresponding to \(g\), and \(\tilde{\mathcal{P}}_{2k}\) corresponding to the conformal metric \(\tilde{g}=u^{\frac{4}{{n}-2{m}}}\cdot {g}\), for \(u\in C^{\infty}_+(M)\), satisfy the relation\N\[\N\mathcal{P}_{2k}(u\phi)=u^{2^*_{{k}}-1}\ \tilde{\mathcal{P}}_{2k}(\phi), \quad 2^*_{k}:=\frac{2{n}}{{n}-2{k}},\N\]\Nfor any \(\phi\in C^{\infty}(M)\). \N\NThe study of these operators leads to an equation analogous to the well-known Yamabe equation. By taking \(\phi\equiv1\), it leads to the \(Q\)-curvature equation:\N\[\N\mathcal{P}_{2k}(u)=\tilde{\mathcal{P}}_{2k}(1)\ |u|^{2^*_{{k}}-2}u.\N\]\NFor any GJMS operator \(P_{2k}\), the function \(Q_{2k}:=\frac{2}{{n}-2{k}}{\mathcal{P}_{2k}(1)}\) is known as the \(Q\)-curvature. Positive solutions to the associated equation yield conformal metrics with prescribed \(Q\)-curvature. The function \(Q_{2k}\) is considered a curvature quantity since the operators \(P_{2k}\) are constructed in terms of covariant derivatives and tensorial operations on the curvature tensor. \N\NRecall that the Yamabe invariant of order \(2k\) is the conformal invariant:\N\[\NY_{2k}:=\inf_{u\in C^{\infty}_+(M)}\frac{\int_M u\ P_{2k}(u)\ dv_g}{\left(\int_M u^{2^*_k}\ dv_g\right)^{\frac{n-2k}{n}}}.\N\]\NGiven a smooth function \(f\) on \(M\), Theorem 1.1 establishes the existence of least energy solutions of class \(2k\) to the equation:\N\[\N\mathcal{P}_{2k}(u)=f\ |u|^{2^*_{{k}}-2}u,\N\]\Nprovided that \(Y_{2k}>0\), there exists a maximal point \(\xi\in M\) where the Laplacian of \(f\) vanishes, the mass of \(P_{2k}\) at \(\xi\) is positive, and a technical condition involving the derivatives of \(f\) and the Weyl tensor is satisfied. If, in addition, the Green's function of \(P_{2k}\) is positive then \(u\) is positive. Therefore, there exist conformal metrics with prescribed \(Q\)-curvature \(2f/(2-2k)\). \N\NThe computation of the GJMS operators is challenging, but an explicit expression is known on Einstein manifolds, as the authors mention. This expression allows for the immediate determination of coercivity when the scalar curvature is positive. Thus, as a corollary, since the coercivity of \(P_{2k}\) is equivalent to \(Y_{2k}>0\), there exist conformal metrics of prescribed \(Q\)-curvature on Einstein metrics with positive scalar curvature. \N\NSuppose that \(Y_{2k}>0\) and the Green's function of \(P_{2k}\) is positive. In Theorem 1.2, the authors obtain the existence of conformal metrics of constant \(Q\)-curvature under the additional assumption that if \(2k+1\leq n\leq 2k+3\) or the manifold is locally conformally flat, then the operator has positive mass at some point.