Intertwining solutions for magnetic relativistic Hartree type equations

We consider the magnetic pseudo-relativistic Schrödinger equation √(-i∇ - A(x))2 + m2u + V(x)u = (Iα ∗ |u|p)|u|p-2u, in ℝN where N ≥ 3, m > 0, V: ℝN → ℝ is an external continuous scalar potential, A: ℝN →N is a continuous vector potential and Iα(x) = cN, α/|x|N-α (x ≠ 0) is a convolution kernel, cN,α > 0 is a constant, 2 ≤ p < 2N/(N-1), (N - 1) p - N < α < N. We assume that A and V are symmetric with respect to a closed subgroup G of the group O(N) of orthogonal linear transformations of ℝN. If for any x ∈ ℝN \ {0}, the cardinality of the G-orbit of x is infinite, then we prove the existence of infinitely many intertwining solutions assuming that A(x) 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.