TITLE: Stationary point of a smooth function $f:\Bbb R^2\to \Bbb R$
QUESTION [13 upvotes]: Let $f \in C^{\infty}(\mathbb{R^{2}})$ be a function such that: $$\lim_{y \to \pm \infty} f(x,y)=+\infty, \ \forall x \in \mathbb{R},$$ and that $$\lim_{x \to \pm \infty} f(x,y)=-\infty, \ \forall y \in \mathbb{R}.$$
With this hypothesis, does a stationary point necessarily exist? I'm looking for either a proof of the existence or a counterexample.  It sounds false to me, but I didn't find a counterexample.

REPLY [5 votes]: I believe this problem can be solved using a version of the Mountain Pass Theorem. Let
$$ m = \min_{y \in \mathbb R} f(0,y)\,, $$
which exists because $\lim_{y \to \pm \infty} f(0,y) = +\infty$. Because of the other limit condition there exist $x_- < 0 < x_+$, such that
$$ f(x_-,0) < m \;\text{and}\; f(x_+,0) < m\,. $$
Now consider the set
$$ \Gamma = \{ \gamma \in C([0,1], \mathbb R^2 \,:\, \gamma(0) = (x_-,0),\, \gamma(1) = (x_+,0) \}\,,$$
of continuous paths between $(x_-,0)$ and $(x_+,0)$. Then a version of the Mountain Pass Theorem should allow us to conclude that
$$ c = \inf_{\gamma \in \Gamma} \max_{t \in [0,1]} \gamma(t)\,,$$
is a critical value of $f$.
There are several formulations of the theorem and I do not have the right formulation at hand, but this question on MathOverflow as well as 
Wikipedia should contain links to the relevant literature.