Semiclassical limit for nonlinear Schrödinger equations with electromagnetic fields

Multiple solutions to a nonlinear Schrödinger equation with Aharonov–Bohm magnetic potential

“…Existence and multiplicity of semiclassical solutions are given e.g. in [3,[6][7][8][9]16]. Very recently, Abatangelo and Terracini obtained existence results for problems with critical nonlinearity and singular magnetic and electric potentials [1].…”

Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

“…A multiplicity result for solutions of (1.3) near global minima of V has been obtained in [16] using topological arguments. A solution that concentrates as → 0 around an arbitrary non-degenerate critical point of V has been obtained in [19] but only for bounded magnetic potentials. Subsequently this result was extended in [20] to cover also degenerate, but topologically non trivial, critical points of V and to handle general unbounded magnetic potentials A.…”

Bound states of nonlinear Schrödinger equations with magnetic fields

“…Thus for equations with the magnetic field it has been studied much less than for that without the magnetic field, see [2,[9][10][11]13,17,20,23,24] and references therein for results on equations with the magnetics field. If the magnetic vector A(x) ≡ 0, it seems that the first work was [17] which studied the existence of solutions of (P 1 ) (i.e., ε = 1) by assuming that σ (− A +V ) ⊂ (0, ∞) where − A := (−i∇ + A) 2 , g(x|u|) = |u| p−2 , N = 2 or 3 and other conditions.…”