Let $w=e^{cu}$, then the equation for $w$ is $$\Delta w-cfw=0,$$ which is a Schrodinger equation, is there is a sufficient condition for Schrodinger equation?

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: $$\lambda_1(-\Delta+cf)\geq 0.$$