Paneitz-Branson invariants on non Einstein manifolds (Q6635317)

The Paneitz-Branson operator on a \((M^n,g)\), an \(n\)-dimensional smooth compact Riemannian manifold with \(n\ge 5\), is defined as \N\[\NP_g^nv=\Delta_g^2v-\text{div}_g(a_nS_gg+b_n\mathrm{Ric}_g)dv+\frac{n-4}{2}Q_g^nv \text{ for } v\in C^4(M^n),\N\]\Nsuch that \(S_g\) (resp. \(\mathrm{Ric}_g\)) is the scalar curvature (resp. the Ricci curvature) of \(g\), \(a_n\) and \(b_n\) are constants given in terms of \(n\), and \(Q_g^n\) is a functions provided in terms of \(n\), \(\Delta_g\), \(S_g\), and \(\mathrm{Ric}_g\). The Paneitz-Branson invariant \(\mu\) is defined as follows: \N\[\N\mu=\inf_{0<u\in C^\infty(M^n)}\left(\frac{\displaystyle\int_{M^n}uP_g^ndv_g}{\left(\displaystyle\int_{M^n}|u|^{2n/(n-4)}dv_g\right)^{(n-4)/n}}\right).\N\]\NThe Paneitz-Branson invariant of high order \(\mu_k\) for \(k\in \mathbb N\setminus\{0\}\) is defined by \N\[\N\mu_k=\inf_{\overline{g}\in[g]}\lambda_k(\overline{g})\left[\int_{M^n}u^{2n/(n-4)}dv_g\right]^{4/n}.\N\]\Nwhere \([g]=\{\overline{g}=u^{4/(n-4)},0<u\in C^\infty(M^n)\}\), and \(\lambda_k(\overline{g})\) is the \(kth\)-eigenvalue of \(P_g^n\). Then the purpose of the authors is to state that if \(Q_g^n\ge 0\), \(S_g\ge 0\), and \(\mu<K^{-1}_0\) (the best constant in the Sobolev embedding of \(H_2^2(M^n)\), the Sobolev space of order two on \(M^n\)), then \(P_g^nu=\mu_1|u|^{8/(n-4)}u\) has positive solutions \(u\in C^\infty(M^n)\) (see Theorem 1.1 for more details). The case when \(1-2^{-4/n}\mu_2K_0>0\) and \(\lambda_1(g)>0\), then the equation \(P_g^n\omega =\mu_2|\omega|^{8/(n-4)}\omega\) has changing-sign solutions \(\omega\in C^4(M^n)\), the case when \(\lambda_1(M^n)<0\), \(\lambda_2(M^n)>0\), and \(Q_g^n\le 0\), then there exist a positive function \(u\in L^{2n/(n-4)}(M^n)\) and \(\omega\in H_2^2(M^n)\) such that \(P_g^n\omega =\mu_2u^{8/(n-4)}\omega\) has changing-sign solutions (Theorem 1.2).