Periodic and Bloch Solutions to a Magnetic Nonlinear Schrodinger Equation

We study the equation (pA) (-i del + A)(2)u + Vu = vertical bar u vertical bar(p-2)u, where A is an element of C(1,alpha) (R(N),R(N)) and V is an element of C(0,alpha) (R(N)) are 2 pi-periodic in each variable, V > 0, and p is an element of (2,2*) with 2* := infinity if N = 2 and 2* := 2N/N-2 if N >= 3. We address two questions: first, the gauge-dependence problem for 2 pi-periodic solutions u : R(N) -> C and second, the multiplicity of Bloch solutions. Unlike the nonperiodic case where problem (pA) is basically independent of A (it is gauge invariant), in the periodic case this is far from being true. Under some assumptions on A we show that, if there exists a one-to-one correspondence between the 2 pi-periodic solutions of (pA) and those of (pA+Z) preserving their absolute value, then z lies in a, subset of measure zero of R(N). We use this fact to show the existence of an uncountable set of Bloch solutions with real quasimomentum.