Let $K\subset \mathbb{R}^2$ be a closed, convex and non-empty set, $F\colon K\to \mathbb{R}^2$ be $\gamma$-strongly monotone and continuous on $K$.
Remind that a function $F$ is $\gamma$-strongly monotone on $K$ iff $$\langle F(x) - F(y), x - y \rangle \ge \gamma \|x-y\|^2, \quad \forall x, y\in K.$$
We can find a function satisfying the above conditions but not Lipschitzian on $K$, for example, $F_0(x, y) = (x - y^2, y+y^3)$.
However, $F_0$ is Lipschitzian on every bounded subset of $K$.
This arises a question:
Is it true that strongly monotone and continuous function on $K$ also Lipschitzian on every bounded subset of $K$?
It seems impossible to prove the property, but finding a counter-example is pretty hard too.
The function defined by $F(x,y)=(2\sqrt{x},y)$ is $1$-strongly monotone on $[0,1]\times[0,1],$ because $2(\sqrt{x}-\sqrt{x'})(x-x')+(y-y')^2=2(x-x')^2/(\sqrt{x}+\sqrt{x'})+(y-y')^2.$ But it's not locally Lipschitz near $0:$ $F(x,0)/x=x^{-1/2}\to\infty$ as $x\to 0^+.$