Higher topological type semiclassical states for Sobolev critical Dirac equations with degenerate potential

TL;DR

This paper constructs multiple high-energy semiclassical states for a Sobolev critical Dirac equation with degenerate potential, using variational methods to analyze concentration phenomena near potential minima.

Contribution

It introduces a variational approach to find an infinite sequence of higher topological type solutions for the critical Dirac equation with degenerate potential.

Findings

Constructed an infinite sequence of solutions with higher energies

Solutions concentrate around local minima of the potential

Applied minimax and penalization methods in the analysis

Abstract

In this paper, we are concerned with semiclassical states to the following Sobolev critical Dirac equation with degenerate potential, \begin{align*} -\textnormal{i} \eps \alpha \cdot \nabla u + a \beta u + V(x) u=|u|^{q-2} u + |u| u \quad \mbox{in} \,\, \R^3, \end{align*} where $u:\mathbb{R}^3\rightarrow \mathbb{C}^4$, $2<q<3$, $\eps>0$ is a small parameter, $a>0$ is a constant, $\alpha=(\alpha_1, \alpha_2, \alpha_3)$, $\alpha_j$ and $\beta$ are $4 \times 4$ Pauli-Dirac matrices. We construct an infinite sequence of higher topological type semiclassical states with higher energies concentrating around the local minimum points of the degenerate potential $V$. The solutions are obtained from a minimax characterization of higher dimensional symmetric linking structure, which correspond to critical points of the underlying energy functional at energy levels where compactness condition breaks down. Our approach is variational, which mainly relies on penalization method and blow-up arguments along with local type Pohozaev identity.