If $u''>0$ in $\mathbf{R}^+$ then $u$ is unbounded? If $u$ is a positive function such that $u''>0$ in the whole $\mathbf{R}^+$ then $u$ is unbounded?
In fact, I know that if $u''>0$ then $u$ is strictly convex. I think that implies $u$ is coercive. I want to prove it.

Consider $g(x)=e^{-x^2}$. Observe that this is a bounded, positive, everywhere differentiable function, and that for sufficiently large $x$, we have $g''(x)>0$. Thus, an appropriate translation of $g(x)$--in particular, some function of the form $g(x+\alpha)$ for some sufficiently large $\alpha$--will give you a counterexample function $u$.
