Multiple complex-valued solutions for nonlinear magnetic Schrodinger equations

TL;DR

This paper proves the existence of multiple complex-valued solutions to nonlinear magnetic Schrödinger equations in the semiclassical limit, with solutions concentrating around a specific domain where the potential attains its minimum.

Contribution

It establishes the existence of multiple solutions with concentration behavior for magnetic Schrödinger equations under Berestycki-Lions conditions, extending previous results to complex-valued solutions.

Findings

Existence of at least cuplength(K)+1 solutions for small 

Solutions concentrate around the set where the potential attains its minimum

Multiple geometrically distinct solutions are found in the semiclassical limit.

Abstract

We study, in the semiclassical limit, the singularly perturbed nonlinear Schr\"odinger equations $$ L^{\hbar}_{A,V} u = f(|u|^2)u \quad \mbox{in } R^N $$ where $N \geq 3$, $L^{\hbar}_{A,V}$ is the Schr\"odinger operator with a magnetic field having source in a $C^1$ vector potential $A$ and a scalar continuous (electric) potential $V$ defined by \begin{equation} L^{\hbar}_{A,V}= -\hbar^2 \Delta-\frac{2\hbar}{i} A \cdot \nabla + |A|^2- \frac{\hbar}{i}\operatorname{div}A + V(x). \end{equation} Here $f$ is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain $\Omega \subset R^N$ such that \[ m_0 \equiv \inf_{x \in \Omega} V(x) < \inf_{x \in \partial \Omega} V(x) \] and we set $K = \{ x \in \Omega \ | \ V(x) = m_0\}$. For $\hbar >0$ small we prove the existence of at least $cuplenght(K) + 1$ geometrically distinct, complex-valued solutions whose modula concentrate around $K$ as $\hbar \to 0$.