Existence of solution for magnetic Schrödinger equation with the Neumann boundary condition.

We consider the nonlinear Schrödinger equation with magnetic field and the Neumann boundary condition: { − ∇ A 2 u + λ u = | u | p − 2 u in Ω , ν ⋅ ∇ A u = 0 on ∂ Ω , where Ω is a boundary domain in R n with a C 1 boundary, ν is the outward normal vector field at x ∈ ∂ Ω , n ≥ 3 , λ > − μ (A) , ( μ (A) is given by (3)), A ∈ C ∞ (Ω ¯ , R n) is a magnetic vector potential. When the exponent is subcritical, 2 < p < 2 ∗ = 2 n n − 2 we can obtain solutions by Nehari manifold. When the exponent is critical, p = 2 ∗ , we can obtain solutions by constrained minimization arguments. [ABSTRACT FROM AUTHOR]