I have the following problem. I need to consider all the conditions in which the following integral may be equal to zero:
Where $p>0$ is a constant. $f$ is a probability density function and $\iota$ is an indicator function (i.e I am truncating the density). $\lambda(\cdot)$ is an increasing function in $\omega$ and $\phi$ is a random variable, so I am solving this integral for any possible realization of $\phi$. Of course there is a trivial solution when $p\phi=\lambda(\omega*)$ for some $\omega*$. But there exist any other case when this is zero which I am not considering?