Boundary layers to a singularly perturbed Klein–Gordon–Maxwell–Proca system on a compact Riemannian manifold with boundary

Abstract

We study the semiclassical limit to a singularly perturbed nonlinear Klein–Gordon–Maxwell–Proca system, with Neumann boundary conditions, on a Riemannian manifold 𝔐{\mathfrak{M}} with boundary. We exhibit examples of manifolds, of arbitrary dimension, on which these systems have a solution which concentrates at a closed submanifold of the boundary of 𝔐{\mathfrak{M}}, forming a positive layer, as the singular perturbation parameter goes to zero. Our results allow supercritical nonlinearities and apply, in particular, to bounded domains in ℝN{\mathbb{R}^{N}}. Similar results are obtained for the more classical electrostatic Klein–Gordon–Maxwell system with appropriate boundary conditions.

Keywords

• electrostatic klein–gordon–maxwell–proca system
• semiclassical limit
• boundary layer
• riemannian manifold with boundary
• supercritical nonlinearity
• lyapunov–schmidt reduction
• 35j60
• 35j20
• 35b40
• 53c80
• 58j32
• 81v10