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Fourth order equations of critical Sobolev growth. Energy function and solutions of bounded energy in the conformally flat case. (English) Zbl 1086.58009
Summary: Given \((M,g)\) a smooth compact Riemannian manifold of dimension \(n \geq 5\), we consider equations like \[ P_g u = u^{2^\sharp-1} \] where \(P_{g}u = \Delta^{2}_{g}u + \alpha\Delta_{g}u + a_{\alpha} u\) is a Paneitz-Branson type operator with constant coefficients \(\alpha\) and \(a_\alpha\), \(u\) is required to be positive, and \(2^\sharp = \frac{2n}{n-4}\) is critical from the Sobolev viewpoint. We define the energy function \(E_m\) as the infimum of \(E(u) = \|u\|^{2^\sharp}_{2^\sharp}\) over the \(u\)’s which are solutions of the above equation. We prove that \(E_m(\alpha) \rightarrow +\infty\) as \(\alpha \rightarrow +\infty\). In particular, for any \(\Lambda > 0\), there exists \(\alpha_0 > 0\) such that for \(\alpha \geq \alpha_0\), the above equation does not have a solution of energy less than or equal to \(\Lambda \).

MSC:
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
35J35 Variational methods for higher-order elliptic equations
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