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

TL;DR

This paper explores natural constraints in variational methods, providing conditions under which constrained critical points are free, and applies these results to establish multiple solutions for superlinear Schrödinger systems on perturbed domains.

Contribution

It introduces a unified approach to natural constraints in variational calculus and applies it to prove solution multiplicity in Schrödinger systems.

Findings

Established sufficient conditions for constrained critical points to be free

Unified various natural constraint conditions in the literature

Proved multiplicity of solutions for superlinear Schrödinger systems

Abstract

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.