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In this paper, we are concerned with semiclassical states to the followingSobolev 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 $4 \times4$ Pauli-Dirac matrices. We construct an infinite sequence of highertopological type semiclassical states with higher energies concentrating aroundthe local minimum points of the degenerate potential $V$. The solutions areobtained from a minimax characterization of higher dimensional symmetriclinking structure, which correspond to critical points of the underlying energyfunctional at energy levels where compactness condition breaks down. Ourapproach is variational, which mainly relies on penalization method and blow-uparguments along with local type Pohozaev identity.

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