Abstract:

Let $D$ be the Dirac operator and $u:S^{N} \rightarrow \Sigma S^{N} $ be a spinor. This article investigates the multiplicity of solutions for $p$-Dirac equations with concave convex nonlinear terms

$\begin{equation*}\label{eq3.26} D_{p} u =\xi |u|^{q-2}u+\eta |u|^{p^*-2}u, \end{equation*}$

where $D_{p} u=:D({|Du|}^{p-2}Du)$, $1. Firstly, the Sobolev embedding $W^{1, p}(S^{N}, \Sigma S^{N}) \hookrightarrow L^{p^*}(S^{N}, \Sigma S^{N})$ loses its compactness because the equation contains a nonlinear term with critical growth. Therefore, in this paper, we utilize the action of an isometry subgroup on the sphere $S^{N} $ to appropriately reduce the function space under consideration, enabling the $Sobolev$ embedding to regain its compactness; Then, using the theory of orthogonal systems, the function space is decomposed, and combined with the Variant Fountain theorem, it is proved that the equation have a series of low-energy weak solutions and a series of high-energy weak solutions; Finally, it is stated that under certain conditions, there are no weak solutions with positive or negative energy for the equation.

Key words: $p$-Dirac equation, variant fountain theorem, group action