Abstract
This paper deals with a nonlinear eigenvalue problem
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 \({p \neq q}\),
has a nontrivial solution.
Similar content being viewed by others
References
Nolder C.A., Ryan J.: p-Dirac operators. Advances in applied Clifford algebras 19(2), 391-402 (2009)
T. Friedrich, Dirac operators in Riemannian geometry. Providence: American Mathematical Society, 2000.
C.A. Nolder, A-Harmonic equations and the Dirac operator. J. Inequal. Appl, 2010.
Y. Lu, G. Bao, Stability of weak solutions to obstacle problem in Clifford analysis. Advances in Difference Equations (1) (2013), 1-11.
B. Zhang, Y. Fu, Weak solutions for A-Dirac equations with variable growth in Clifford analysis. Electronic Journal of Differential Equations (227) (2012), 1-10.
Fu Yongqiang, Zhang Binlin, Weak solutions for elliptic systems with variable growth in Clifford analysis. Czechoslovak Mathematical Journal, accepted.
X. Fan, Q. Zhang, D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem. Journal of Mathematical Analysis and Applications 302(2) (2005), 306-317.
X. Fan, Remarks on eigenvalue problems involving the p(x)-Laplacian. Journal of Mathematical Analysis and Applications 352(1) (2009), 85-98.
J.E. Gilbert, M. Murray, Clifford algebras and Dirac operators in harmonic analysis. Cambridge University Press, 1991.
P. Lindqvist, Addendum: On the equation div(|∇u|p−2 ∇u) + λ |u|p−2 u = 0. Proc. Amer. Math. Soc. 116 (1992), 583-584.
M.M. Rao, Z.D. Ren, Theory of Orlicz spaces. Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991.
E. Zeidler, The Ljusternik Schnirelman theory for indefinite and not necessarily odd nonlinear operators and its applications. Nonlinear Anal. 4 (1980), 451-489.
E. Zeidler, Nonlinear Functional Analysis and its Applications. Vol.3, Variational Methods and Optimization, Springer, Berlin, 1985.
Le A.: Eigenvalue problems for the p-Laplacian. Nonlinear Analysis: Theory, Methods Applications 64(5), 1057-1099 (2006)
K. Yosida, Functional analysis, (6th edn). Springer-Verlag, Berlin, 1980.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pan, L., Bao, G. On a Eigenvalue Problem Involving Dirac Operator. Adv. Appl. Clifford Algebras 25, 415–424 (2015). https://doi.org/10.1007/s00006-014-0497-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-014-0497-6