Multiplicity and concentration of solutions for Kirchhoff equations with magnetic field (Q1983640)

The authors study a nonlinear magnetic Kirchhoff equation of the form \[ \begin{cases} -(a \varepsilon^2+b\varepsilon[u]^2_{A/\varepsilon})\Delta_{A/\varepsilon}u+V(x)u=f(|u|^2)u \text{ in } \mathbb{R}^3,\\ u\in H^1(\mathbb{R}^3,\mathbb{C}), \, \varepsilon >0, \, a,b>0 \text{ (constants)}. \end{cases} \] In this problem, \(-\Delta_{A}u= \left(\frac{1}{i}\nabla-A(x)\right)^2u\) is the magnetic Laplace operator, \([u]^2_{A}=\int_{\mathbb{R}^3}|\nabla_A u|^2dx\), \(A:\mathbb{R}^3\to\mathbb{R}^3\) is the magnetic potential, \(V:\mathbb{R}^3\to\mathbb{R}\) and \(f \in C(\mathbb{R},\mathbb{R})\) satisfy certain regularities. Combining variational methods, a penalization technique and the Ljusternik-Schnirelmann theory, the authors establish multiplicity properties of solutions and concentration phenomena for \(\varepsilon\) small.