We analyze the asymptotic pointwise behavior of families of solutions to the high-order critical equation $$P_\alpha u_\alpha=\Delta_g^k u_\alpha+\hbox{ lot}=|u_\alpha|^{ 2^\star-2-\epsilon_\alpha} u_\alpha\hbox{ in }M$$ that behave like $$u_\alpha=u_0+B_\alpha+o(1)\hbox{ in }H_k^2(M)$$ where $B=(B_\alpha)_\alpha$ is a Bubble, also called a Peak. We give obstructions for such a concentration to occur: depending on the dimension, they involve the mass of the associated Green's function or the difference between $P_\alpha$ and the conformally invariant GJMS operator. The bulk of this analysis is the proof of the pointwise control \begin{ equation*} |u_\alpha(x)|\leq C\Vert u_0\Vert_\infty^{ (2^\star-1)^2}+C\left(\frac{ \mu_\alpha^{ 2}}{ \mu_\alpha^{ 2 }+d_g(x,x_\alpha)^{ 2 }}\right)^{ \frac{ n-2k}{ 2}}\hbox{ for all }x\in M\hbox{ and }\alpha\in\mathbb{ N}, \end{ equation*} where $|u_\alpha(x_\alpha)|=\max_M|u_\alpha|\to +\infty$ and $\mu_\alpha:=|u_\alpha(x_\alpha)|^{ -\frac{ 2}{ n-2k}}$. The key to obtain this estimate is a sharp control of the Green's function for elliptic operators involving a Hardy potential.
Categories: math.AP