A remark on natural constraints in variational methods and an application to superlinear Schrödinger systems

For a C2-functional J defined on a Hilbert space X, we consider the set , where A⊂X is open and Vx⊂X is a closed linear subspace, possibly depending on x∈A. 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 Schrödinger systems on singularly perturbed domains.