Nonhomogeneous Eigenvalue Problem With Indefinite Weight

The eigenvalue problem is a very important research content in the field of partial differential equations.In recent years,many mathematicians have paid attention to the eigenvalue problem of nonlinear equations,especially for the eigenvalue problem of variable exponent partial differential equations.This paper,we will consider the distribution of eigenvalue of the following equation based on the theory of the Orlicz-Sobolev spaces and exponent Lebesgue spaces(?)Where ?(?)R~N(N?3),function a:(0,?)?R,we suppose that ?>0,V is an indefinite sign-chaging weight and q:??(1,?)is a continuous function.In this paper,the research methods of nonlinear analysis such as Weierstrass theorem are used to have the following results.(1)The minimum positive eigenvalue X and the maximum negative eigenvalue ?~*are obtained;(2)Any ??(-?,?)?(?~*,+?)is an eigenvalue of problem(P)while any??(?~*,?~*)contains no eigenvalue.