Solving nonlinear non-local problems using positive square-root operators

A non-constructive existence theory for certain operator equations Lu = Du, using the substitution u = B ½ξ with B = L −1, is developed, where L is a linear operator (in a suitable Banach space) and D is a homogeneous nonlinear operator such that Dλu = λα Du for all λ ≥ 0 and some α ∈ ℝ, α ≠ 1. This theory is based on the positive-operator approach of Krasnosel’skii. The method has the advantage of being able to tackle the nonlinear right-hand side D in cases where conventional operator techniques fail. By placing the requirement that the operator B must have a positive square root, it is possible to avoid the usual regularity condition on either the mapping D or its Fréchet derivative. The technique can be applied in the case of elliptic PDE problems, and we show the existence of solitary waves for a generalization of Benjamin’s fluid dynamics problem.