Summary: We establish some minimax theorems, of purely topological nature, that, through the variational methods, can be usefully applied to nonlinear differential equations. Here is a (simplified) sample: Let \(X\) be a Hausdorff topological space, \(I\subseteq\mathbb R\) an interval and \(\Psi:X\times I\to]-\infty,+\infty]\). Assume that the function \(\Psi(x,\cdot)\) is lower semicontinuous and quasi-concave in \(I\) for all \(x\in X\), while the function \(\Psi(\cdot,q)\) has compact sublevel sets and one local minimum at most for each \(q\) in a dense subset of \(I\). Then, one has \[ \sup_{q\in I} \inf_{x\in X} \Psi(x,q)= \inf_{x\in X} \sum_{q\in I} \Psi(x,q). \]