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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
##### Keywords:
Paneitz-Branson type operator
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