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?