A standard trick is to formulate your problem in such a way that your solution candidates are in some bounded region in a space to which Banach-Alaoglu applies, then introduce some kind of approximating sequence and use Banach-Alaoglu to extract a weak-star convergent subsequence. (The approximation scheme depends on the problem; for example, you could introduce an artificial regularization term, solve the regularized problems, and send the regularization to zero). The hard part is usually justifying that this weak-star limit has anything to do with the problem you started with.