On a Eigenvalue Problem Involving Dirac Operator

Abstract

This paper deals with a nonlinear eigenvalue problem

$$\left\{\begin{array}{ll}D(|Du|^{p-2}Du) + \lambda|u|^{p-2}u = 0,\, \, \, \, x \in \Omega; \\ u = 0, \qquad\qquad\qquad\qquad\quad {\rm on}\, \partial \Omega. \end{array}\right .$$

where ${\Omega}$ is a bounded domain with a piecewise smooth boundary in R ⁿ, D is the Euclidean Dirac operator. We prove that this problem has at least one sequence of solutions (${u_n, \lambda_n}$) such that ||u _n || _p =  1. Here ${{\{\lambda_n\}}}$ is a nondecreasing sequence which tends to ${\infty {\rm as} n \rightarrow \infty}$. The eigenvalues for this problem have a positive lower bound, and the spectrum is a closed set. We also show that when ${p \neq q}$,

$$D(|Du|^{p-2}Du) + |u|^{q-2}u = 0,$$

has a nontrivial solution.