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I wonder if there is any paper on the "priori estimate approach" for Yamabe problem.

The Yamabe problem is solving the following equation: On a $C^{\infty}$ compact Riemannian manifold $M_n$ of dimension $n \geq 3$, let us consider the differential equation

$$ \Delta \varphi+h(x) \varphi=\lambda f(x) \varphi^{N-1} $$

where $h(x)$ and $f(x)$ are $C^{\infty}$ functions on $M_n$, with $f(x)$ everywhere strictly positive and $N=2 n /(n-2)$.

As a graduate student, the materials I read are all based on variational method, did any scholars have studied the Yamabe problem through priori estimate approach? For example, the $C^k$ or $W^{k,p}$ estimate by maximum principal or other tools.

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    $\begingroup$ The variational approach does require a priori estimates. Another approach commonly used to prove the existence of solutions to a nonlinear PDE is the continuity method. For solutions that don’t minimize the Yamabe functional, one could try Morse theoretic approaches. All of the approaches require a priori estimates. $\endgroup$
    – Deane Yang
    Commented Nov 6 at 16:21
  • $\begingroup$ Thanks for your comment! maybe I didn't express my question clearly or my question is too naive, I mean that is there papers trying to solve the equation without treating the solution as the critical point of the functional, for example, using the continuity method you mentioned (the estimate for each order of the solution). $\endgroup$
    – Elio Li
    Commented Nov 7 at 5:56
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    $\begingroup$ There is also the heat flow approach, which was actually carried out by Brendle. en.wikipedia.org/wiki/Yamabe_flow $\endgroup$
    – Deane Yang
    Commented Nov 7 at 12:47

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