Higher Topological Type Semiclassical States for Sobolev Critical Dirac Equations with Degenerate Potential

Abstract

In this paper, we are concerned with semiclassical states to the following Sobolev critical Dirac equation with degenerate potential, $$-\text {i} \epsilon \alpha \cdot \nabla u + a \beta u + V(x) u=|u|^{q-2} u + |u| u \quad \text{ in } \,\, \mathbb {R}^3, $$ where $$u:\mathbb {R}^3\rightarrow \mathbb {C}^4$$ , $$2<q<3$$ , $$\epsilon >0$$ 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. Here the degeneracy of V means that $$|V(x)|<a$$ for any $$x \in \mathbb {R}^3$$ and |V(x)| may approach a as |x| tends to infinity. 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.