Let us first recall a classical theorem for saddle points.
Suppose that $X,Y$ are two reflexive Banach spaces, $E\subset X$ and $F\subset Y$ are two non-empty closed convex sets. Assume that
(1) For any $y\in F$, the mapping $x\to f(x,y)$ is l.s.c. and quasi-convex
(2) For any $x\in E$, the mapping $y\to -f(x,y)$ is l.s.c. and quasi-convex.
(3) There exits some $(x_0,y_0)\in E\times F$ such that $f(x_0,y)\to -\infty$ as $\|y\|\to +\infty$ and $f(x,y_0)\to\infty$ as $\|x\|\to+\infty$.
Then there exits a saddle point $(x^*,y^*)\in E\times F$ such that $$ \max_{y\in F}f(x^*,y)=f(x^*,y^*)= \min_{x\in E} f(x,y^*) \quad (A) $$
I know that if $E$ and $F$ are weakly compact, then the condition about can be written as $$ \min_{x\in E}\max_{y\in F}f(x,y)=\max_{y\in F}\min_{x\in E}f(x,y). \quad (B) $$ I'm interesting in find an counter-example that the functional satisfies (1)-(3) and hence $(A)$ but not $(B)$. The key part seem to allow $E$ and $F$ to be unbounded.