Intertwining solutions for magnetic relativistic Hartree type equations. (16th April 2018) - Cingolani, Silvia; Secchi, Simone
Abstract: We consider the magnetic pseudo-relativistic Schrödinger equation where, m > 0, is an external continuous scalar potential, is a continuous vector potential and is a convolution kernel, is a constant, , . We assume that A and V are symmetric with respect to a closed subgroup G of the group of orthogonal linear transformations of . If for any, the cardinality of the G -orbit of x is infinite, then we prove the existence of infinitely many intertwining solutions assuming that is either linear in x or uniformly bounded. The results are proved by means of a new local realization of the square root of the magnetic laplacian to a local elliptic operator with Neumann boundary condition on a half-space. Moreover we derive an existence result of a ground state intertwining solution for bounded vector potentials, if G admits a finite orbit.