I am looking for a function $u(x) : R^+ \to R$, such that
$\frac{w}{1+p}$ is a solution to $\max_x u(x) \text{ s.t. } px \leq w$.
Is there some continuous function for which this is the case? I need it for a possible counterexample, or to find out more about what properties of $u(x)$ might be useful for a proof I am working on.
If you want (economic) background: I am looking for the utility function that induces said demand function. I want to show that $\pi(p)=p \cdot x^*(p,w)$ has or doesn't have a maximum if $x^*(p,w)$ is the solution to the above maximization problem, where $u(x) : R^+ \to R$ is a continuos, strictly quasi-concave function. Of course, $ \frac{wp}{1+p}$ has no maximum, that's why I am interested in the generating utility function.