Solvability of quasilinear elliptic equations on closed manifolds

Question

Is there any reference about solvability theory of quasilinear elliptic equations on closed manifolds?

In particular, I am looking for solvability condition for function $f$ of following equation $\Delta u+ c |\nabla u|^2=f$ on a closed manifold $M$, $c$ is a positive constant. As far as I understand $\int_M\, \Delta u+c |\nabla u|^2\, dV=\int_M\, c |\nabla u|^2\, dV\geq 0$, so at least $\int_M\, f\, dV\geq 0$, is that enough? Is there any sufficient condition for solvability?

Thanks a lot.

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Now seems like the statement of the problem can be simplified as following

Under what condition for $f$, there exists positive solution $u$ for the Schrodinger equation $\Delta u+f u=0$ on a closed manifold.

Answer 1

No. Consider the case of periodic solutions in one dimension, and let c=1. That is, we have the equation $$u''+(u')^2=f.$$ Multiply both sides by $e^u$. You end up with $$(e^u)''=e^uf.$$ Now if, for instance, $f$ is positive, then the right hand side is positive, and the left hand side has zero average, which cannot be.

Answer 2

Let $w=e^{cu}$, then the equation for $w$ is $$\Delta w-cfw=0,$$ which is a Schrodinger equation. A necessary for a Schrodinger equation to have a positive solution is Barta's lemma: $$\lambda_1(-\Delta+cf)\geq 0.$$