For a C 2-functional J defined on a Hilbert space X, we consider the set N={x∈A:projVx∇;J(x)=0}, where A⊂X is open and V x⊂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
Noris, B., Verzini, G. (2012). A remark on natural constraints in variational methods and an application to superlinear Schrodinger systems. JOURNAL OF DIFFERENTIAL EQUATIONS, 254(3), 1529-1547 [10.1016/j.jde.2012.11.003].
A remark on natural constraints in variational methods and an application to superlinear Schrodinger systems
NORIS, BENEDETTA;
2012
Abstract
For a C 2-functional J defined on a Hilbert space X, we consider the set N={x∈A:projVx∇;J(x)=0}, where A⊂X is open and V x⊂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
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