The Solution Of A Nonlinear Schr(?)dinger Equation With A Magnetic Field On A Bounded Domain

In this paper, we deal with the equationIt is physically meaningful. Similar equation models are involved in superconductivity, Bose-Einstein condensation and liquid crystal problem. There already have been lots of results about equation without A and linear equation with A. We are concerned with the existence and the regularity of the weak solution to the nonlinear Schrodinger equation with a magnetic field in this paper. The main result includes two parts:(1)Existence: Suppose A ∈ L^q(Ω), q≥4, μ(A) > 0, f satisfy (f1) - (f3); or A∈L^∞(Ω), μ(A) > 0, f satisfy (f1), (f2), (f4). If one of the conditions above is satisfied, then there exists a weak solution to the equation.(2)Regularity: (a)Suppose Ω. is a bounded smooth domain, (?)Ω is a C² curve. A∈L^q(Ω), q ≥ 3, DA ∈ L^∞(Ω), f'(t) ≤ t^(p-3)/2, if u is a weak solution to the equation, then u ∈ H²(Ω).(b)Suppose Ω is a bounded smooth domain, (?)Ω is a C³ curve. A ∈ L^q(Ω), q ≥ 3, DA ∈ L^∞(Ω,), f"(t) ≤ t^(p-5)/2, if u is a weak solution to the equation, then u ∈ H³(Ω).