Semiclassical states for NLS equations with magnetic potentials having polynomial growths

In this paper, the authors consider in a semiclassical regime a nonlinear Schrödinger equation in R3 with both an electric and a magnetic field. More precisely, the nonlinear Schrödinger equation under study is of the form (−iℏ∇−A)2u+Vu=f(|u|2)u, where A is a vector potential for the magnetic field, V is a strictly positive electric potential, f:[0,∞)→R is an increasing differentiable function and u:R3→R. The main result of this work states that if A, V and f satisfy some conditions, which are too technical to be stated here, then there exists ℏ0>0 such that for each ℏ with 0<ℏ<ℏ0 there exists a solution u of the above NLS equation that satisfies ∫R3∣∣[(−iℏ∇−A)u](x)∣∣2dx+∫R3V(x)∣∣u(x)∣∣2dx<∞. This statement greatly extends some of the results already contained in previous work of the same authors [J. Math. Anal. Appl. 275 (2002), no. 1, 108--130; MR1941775 (2003k:81062)]. In particular, the boundedness assumption on the electric field and on the vector potential is removed. As pointed out in this paper, the physically meaningful case of a constant magnetic field is now included in the present framework.