Let $\mathbb {\overline B}^n\subset \mathbb R^n$ be a closed unit ball in and let $\mathbb B^k\subset \mathbb R^k$ be a open unit ball.

Suppose $F$ is a smooth function on $\mathbb {\overline B}^n\times \mathbb B^k$ that has the following properties.

1. $F(x,y)$ tends to $-\infty$ when $|y|\to 1$ for $(x,y)\in \mathbb {\overline B}^n\times \mathbb B^k$.

2. For any $y_0\in \mathbb B^k$ the strict minimum of $F$ on the ball $\mathbb {\overline B}^n\times y_0$ is attained in the interior of the ball. (Here by strict I mean, that the minimum of $F$ on the boundary of $\mathbb {\overline B}^n\times y_0$ is larger than the minimum of $F$ over the whole ball.)

Question. Is it true that $F$ has a critical point $(x_0,y_0)$ in $\mathbb {\overline B}^n\times \mathbb B^k$ such that $F$ restricted on $\mathbb {\overline B}^n\times y_0$ has a minimum at $(x_0,y_0)$? If yes, I would like to know if there are some natural generalizations of this statement. If no, could one impose some additional conditions that guarantee a positive answer?