Function with infinitely many maxima and no minima

Question

I'm looking for an example of a function $f\in C^\infty$, $f:\mathbf{R}^2\to(0,\infty)$, $\lim_{\|x\|\to\infty}f(x)=0$ such that is has infinitely many local maxima, infinitely many saddle point and has no local minima. I have some intuitive view how such function may look like, but apart from some intuitions I don't know exactly what this function could be.

Answer 1

How about

$$ f(x,y)=\frac{\cos(x)+2}{1+x^2+y^2}$$