  Hamiltonian Floer theory plays an important role for finding periodic
solutions of Hamilton's equation, which can be seen as a generalization of
Newton's equation. Generalizing Newton's equation to Laplace's equation with
non-linearity, we show, building on the work of Ginzburg and Hein, that this
role is taken over by the hyperk\"ahler Floer theory of Hohloch, Noetzel, and
Salamon. Apart from establishing $C^0$-bounds in order to be able to deal with
noncompact hyperk\"ahler manifolds, the core ingredient is a regularization
scheme for the polysymplectic formalism due to Bridges, which allows us to link
Euclidean field theory with hyperk\"ahler Floer theory. As a concrete result,
we prove a cuplength estimate.
