An integro-differential inequality related to the smallest positive eigenvalue of p(x)-Laplacian Dirichlet problem

Abstract

We consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem. 1. Introduction In recent years there has been an increasing interest in the study of various mathematical problems with variable exponent (see for example [1, 11, 12, 14, 17]). Differential equations and variational problems with p(x)-growth conditions arise from the study of elastic mechanics, oscillation problem, electrorheological fluids or image restoration ([5, 6, 16]). The basic properties of variable exponent function spaces were derived by O. Kováčik and J. Rákosník in [13] and (by different methods) by X.-L. Fan and D. Zhao in [10]. For a comprehensive survey concerning Lebesgue and Sobolev spaces with variable exponent we refer to [7]. One of the most interesting topics is the p(x)-Laplacian Dirichlet problem. Our interest is in the smallest eigenvalue of this problem. X. Fan, Q. Zhang and D. Zhao in [9] studied the following eigenvalue problem (−div(|∇u|p(x)−2∇u) = λ|u|p(x)−2u in D, u = 0 on ∂D, where D is a bounded domain in Rn, p: D →(1, ∞) is a continuous function and λ is a real number. Denoting by Λ the set of all nonnegative eigenvalues, they