Multi-peak solutions for magnetic NLS equations without non--degeneracy conditions

TL;DR

This paper proves the existence of multi-peak solutions for a magnetic nonlinear Schrödinger equation with complex potentials and nonlinearities, without relying on non-degeneracy conditions, advancing understanding of such equations in quantum physics.

Contribution

It establishes the existence of semiclassical multi-peak solutions for magnetic NLS equations under nearly optimal conditions, without requiring non-degeneracy assumptions.

Findings

Existence of multi-peak solutions in magnetic NLS equations.

Solutions can be constructed even with unbounded magnetic potentials.

Results apply to equations with multi-well and vanishing electric potentials.

Abstract

In the work we consider the magnetic NLS equation (\frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u = 0 \quad {in} \R^N where $N \geq 3$, $A \colon \R^N \to \R^N$ is a magnetic potential, possibly unbounded, $V \colon \R^N \to \R$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u\colon \R^N \to \C$, under conditions on the nonlinearity which are nearly optimal.