A remark on natural constraints in variational methods and an application to superlinear Schr\"odinger systems

For a regular functional J defined on a Hilbert space X, we consider the set N of points x of X such that the projection of the gradient of J at x onto a closed linear subspace V(x) of X vanishes. We study sufficient conditions for a constrained critical point of J restricted to N to be a free critical point of J, providing a unified approach to different natural constraints known in the literature, such as the Birkhoff-Hestenes natural isoperimetric conditions and the Nehari manifold. As an application, we prove multiplicity of solutions to a class of superlinear Schr\"odinger systems on singularly perturbed domains.