Nonlinear Schrödinger Equations with Steep Magnetic Well

Abstract

We study the nonlinear magnetic Schrödinger equation, $-(\nabla -i\lambda A)^{2}u=f(x,|u|^{2})u$ on $\mathbb{R}^{N}$, where $N \geq 2$ and the nonlinearity is super-linear and subcritical. The vector potential $A$ and the associated magnetic field are assumed to vanish on a common bounded open set $\Omega$. It is shown that the equation above has more and more solutions which are localized near $\Omega$ as $\lambda \to \infty$.