Semiclassical states for a static supercritical Klein–Gordon–Maxwell–Proca system on a closed Riemannian manifold

Abstract

We establish the existence of semiclassical states for a nonlinear Klein–Gordon–Maxwell–Proca system in static form, with Proca mass 1, on a closed Riemannian manifold. Our results include manifolds of arbitrary dimension and allow supercritical nonlinearities. In particular, we exhibit a large class of three-dimensional manifolds on which the system has semiclassical solutions for every exponent p>2. The solutions we obtain concentrate at closed submanifolds of positive dimension as the singular perturbation parameter goes to zero

Klein-Gordon-Maxwell equations in high dimensions

“…Blow-up solutions for equations like (9.9) can be found in Esposito, Pistoia and Vétois [30], Hebey [35], and Robert and Vétois [48,47]. We refer also to Druet, Hebey and Vétois [27], and Hebey and Wei [39] on what concerns (0.1), and to Clapp, Ghimenti and Micheletti [14] and Ghimenti and Micheletti [32] for the semiclassical setting associated to (0.1).…”

Vortex ground states for Klein-Gordon-Maxwell-Proca type systems

“…Existence and non-existence results, under different assumptions on the nonlinearity and on m and ω are present in [1,10,11], while the existence of a ground state has been considered in [2,30]. Moreover we mention [13,14], for the bounded domain case, and [9,15,17,19,20], for the case A = 0 and µ = 0 on manifolds. Finally Klein-Gordon equations coupled with Born-Infeld type equations have been treated in [8,12,32].…”

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

“…Thereafter several works are devoted to the study of KGMP system on Riemaniann closed manifold. We limit ourself to cite [18,19,5,16,17]. Klein Gordon Maxwell system provides a model for a particle u interacting with its own electrostatic field v. Thus, is somewhat more natural to prescribe Neumann condition on the second equation as d'Avenia Pisani and Siciliano nicely explained in the introduction of [8].…”