Nonexistence of solutions in $(0,1)$ for K-P-P-type equations for all $d\ge 1$

Consider the KPP-type equation of the form $\mathrm{\Delta }u+f(u)=0$, where $f:[0,1] \to \mathbb R_{+}$ is a concave function. We prove for arbitrary dimensions that there is no solution bounded in $(0,1)$. The significance of this result from the point of view of probability theory is also discussed.