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Let $(M,g)$ be a closed, Riemannian manifold of dimension greater than two. Let $u$ a positive solution of the equation

$\Delta u - c u = -du^\frac{n+2}{n-2}$,

where $\Delta = -div\nabla$ and $c$ and $d$ are positive constants. I've read that a consequence of the maximum principle is that $u$ will be the unique non-trivial solution but I can't find a proof anywhere. Does someone know how to prove this assertion?

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    Willie Wong: I've encountered this result in a few papers about the Yamabe problem. It is stated as a fact in a paper by Richard Schoen (I forgot the title, but it has something to do with counting the number of constant scalar curvature metrics in a conformal class.) The specific statement is that there is one smooth positive solution to the Yamabe equation when the Yamabe constant is negative.2011-06-28

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I figured it out. Actually the method is not so much the maximum principle as it is a type of maximum principle. One just writes: $\Delta u = -d u^{\frac{n+2}{n-2}} + c u$ and checks what constraints on $u$ must be satisfied at a min (where $\Delta u < 0$) and at a max (where $\Delta u < 0$). The inequalities that you get rule out the possibility of a positive solution that is not constant. It still is possible, though, for a solution to exist that is negative somewhere.