I found some notes where are dedicated to find bounded solutions for certain pde. In these notes, they defined the energy $E$ and the angular momentum $J$, and they consider solutions of the form $r(x)\exp(\theta (x))$. For these ansatz, they write the energy $E=\frac{(r')ˆ2}{2} + V(r)$, where $V$ is called the effective potential.
Later, they say since $V'(r)$ is negative, then there is no bounded solutions. I don't understand that. At first, I thought it has to do with Lyapunov function... but this is used to study the stability of the solution and not the existence I think... So what is the author using when they say since $V'(r)$ is negative, then there is no bounded solutions?
Thanks in advance
EDIT: $V$ has this form $V(r)=Jˆ2/(2rˆ2)+(rˆ2)/2 - (rˆ4)/4$ and $V'$ is negative whenever $J$ is greater than some constant.