Spectrum of the Ap-Laplacian Operator

Abstract

This work deals with the nonlinear boundary eigenvalue problem(V:P(Gammaho;I)):-A_p u = lambda ho(x)|u|^{p-2}u in I =], b[,u(a) = u(b) = 0,where A_p is called the A_p-Laplacian operator and defined by A_p u = (Gamma(x) |u'|^{p-2}u'),p > 1, lambda is a real parameter, ho is an indefinite weight, a, b are real numbers and Gamma in C^1(I) cap C^0(overline{I}) and it is nonnegative on overline{I}.We prove in this paper that the spectrum of the A_p-Laplacian operator is given by a sequence of eigenvalues. Moreover, each eigenvalue is simple, isolated andverifies the strict monotonicity property with respect to the weight ho and the domain I. The k¡th eigenfunction corresponding to the k-th eigenvalue has exactly k-1 zeros in (a,b). Finally, we give a simple variational formulation of eigenvalues.

Keywords

• A_p-Laplacian spectrum
• Nonlinear eigenvalue problem.