Suppose $U \in \mathbb R^n$ is an open simply connected set and $f: U \to \mathbb R$ is a real valued $C^{\infty}$ function. I am wondering whether the following is possible: $f$ has more than $1$ local minimizers, say $x_1, x_2 \in U$ but does not have any other saddle points or local maximizers.
I believe if $n=1$ this cannot happen but not sure whether things change in higher dimensions.
You can construct a function $f:\mathbb R^2\to\mathbb R$ with local minima at $(\pm1,0)$ and a saddle point at $(0,0)$, and no other critical points, such that $f(x,y)\to+\infty$ as $x^2+y^2\to\infty$. Now let $U$ equal $\mathbb R^2\setminus A$, where $A$ all points in the plane within distance $\le \epsilon$ of the positive $y$ axis, for some small $\epsilon$. The set $U$ is simply connected. But $f$ has two local minima in $U$, and no other critical points.
I don't really understand what you are asking for, so I don't know if this is a counterexample, or what. I offer it as an invitation for you to clarify your question.
