Eigenvalue Problems And Picone's Identity For P(x)-laplace Equations

In the past 20 years,the differential equations and variational problems with non-standard growth conditions and the corresponding function spaces with variable exponents have been at-tractive for the researchers in these fields.These investigations are mainly stimulated by the development of the study of electrorheological fluids,image restoration and the theory of non-linear elasticity.In this thesis,the framework is closely related to this subject,and the existence and stability of the eigenvalues for p(x)-Laplace equation with Robin boundary condition are s-tudied.Moreover,in order to study the property of the first eigenvalue for p(x)-Laplace equation subject to Dirichlet boundary condition,Picone's identity of p(x)-Laplace equation is proved,and several applications of this formula to p(x)-Laplace equation are discussed.This thesis is organized as follows.In Chapter 1,the background,research status and main results are given.In Chapter 2,the preliminaries are recalled.Chapter 3 concerns the existence of infinitely many eigenvalues for Robin eigenvalue prob-lem involving p(x)-Laplace operator,and establishes the Euler-Lagrange equation for the min-imization of a Rayleigh quotient of two Luxemburg norms in the framework of variable expo-nent Sobolev space.Notice that this Rayleigh quotient is homogeneous of degree 1.Finally,our main aim is to prove the existence of infinitely many eigenvalues and also show that,the smallest eigenvalue is strictly positive,and all eigenfunctions associated with this eigenvalue do not change sign.Chapter 4 discusses the stability of eigenvalues for p(x)-Laplace equation subject to Robin boundary condition in the framework of ?-convergence.The purpose is to prove that the unifor-m convergence of the exponents is enough to guarantee the convergence of the m-th eigenvalues for Robin problem,which can be defined by inf sup sequences of Rayleigh quotient involving Luxemburg norms.Chapter 5 is devoted to presenting a Picone-type identity for p(x)-Laplace equation and establishing several applications of this formula to p(x)-Laplace equation,such as Caccioppoli inequalities,nonexistence of positive supersolutions,domain monotonicity property,unique-ness and simplicity of the first eigenvalue,Hardy type inequality,Barta type inequality,linear relationship between the components of the solution to a nonlinear singular elliptic system and Sturmian comparison theorem.