Low energy solutions for singularly perturbed coupled nonlinear systems on a Riemannian manifold with boundary

Abstract

Abstract. Let (M, g) be a smooth, compact Riemannian manifold with smooth boundary, with n = dim M = 2, 3. We suppose the boundary ∂M to be a smooth submanifold of M with dimension n − 1. We consider a singularly perturbed nonlinear system, namely Klein-Gordon-Maxwell-Proca system, or Klein-Gordon-Maxwell system of Scrhoedinger-Maxwell system on M . We prove that the number of low energy solutions, when the perturbation parameter is small, depends on the topological properties of the boundary ∂M , by means of the Lusternik Schnirelmann category. Also, these solutions have a unique maximum point that lies on the boundary.

Schrödinger–Maxwell systems on compact Riemannian manifolds

“…In particular, this research stream reached as well the study of Schrödinger-Maxwell systems. Indeed, in the last five years Schrödinger-Maxwell systems has been studied on n−dimensional compact Riemannian manifolds (2 ≤ n ≤ 5) by Druet and Hebey [11], Hebey and Wei [15], Ghimenti and Micheletti [12,13] and Thizy [29,30]. More precisely, in the aforementioned papers various forms of the system…”

Klein–Gordon–Maxwell Systems with Nonconstant Coupling Coefficient

“…Most results in the literature concern systems posed in unbounded spatial domains, possibly featuring lower-order nonlinear perturbations; for instance, see [3,7,8,14,17]. We also refer to [12,13] for recent applications to Klein-Gordon-Maxwell systems with Neumann boundary conditions on Riemannian manifolds.…”

Nonlinear Klein-Gordon-Maxwell systems with Neumann boundary conditions on a Riemannian manifold with boundary

“…So, recently we moved to study KGMP systems in a Riemaniann manifold M with boundary ∂M with Neumann boundary condition on the second equation. In [14] the authors proved that the topological properties of the boundary ∂M , namely the Lusternik Schnirelmann category of the boundary, affects the number of the low energy solution for the systems. Also, we notice that the natural dimension for KGM and KGMP systems is n = 3, since this systems arises from a physical model.…”

“…However, the case n = 4 is interesting from a mathematical point of view, since the second equation of systems ( 1) and ( 2) becomes energy critical by the presence of the u 2 v term. For further comments on this subject, we refer to [18] We can compare [14] and Theorem 2. In [15] we proved that the set of metrics for which the mean curvature has only nondegenerate critical points is an open dense set among all the C k metrics on M , k ≥ 3.…”