Existence And Multiplicity Of Solutions For A Neumann Problem Involving The P(x)-Laplace Operator

The p(x)-Laplacian equation is a basic equation in nonlinear elasticity theory. More recently the p(x)-Laplacian systems have attracted more and more attention be-cause they can model phenomena which arise from the study of electrorheolgical fluids or elastic mechanics. Comparing with the p-Laplacian operator, the p(x)-Laplacian operator possesses more complicated nonlinear properties. This causes many prob-lems and some classical theories and methods are not applicable. The main methods of study the p(x)-Laplacian systems is the three critical point theorem due to Ric-ceri. This theorem basically solve the existence and multiplicity of the p(x)-Laplacian system and have done a lot of promotion by domestic and foreign scholars.This article is a summary about the existence and multiplicity of the following equation under different conditions. Since the 1980s, Pro. Xian-ling Fan, Mihai Mihailescu and some other professors make use of the following theorems and the definition of the weak solutions, then they get the existence of the Neumann problem involving the p(x)-Laplace operator.Theorem 1. Let X be a separable and reflexive real Banach space;Φ:x→R a contin-uously Gateaux differentiable and sequentially weakly lower semicontinuous fuctional whose Gateaux derivative admits a continuous inverse on X*;Ψ:X→R a continu-ously Gateaux differentiable functional whose Gateaux derivative is compact. Assume that Then there exist on an open interval A (?) (0,∞) and a positive real number q such that for eachλ∈∧the equationΦ'+λΨ'= 0 has at least three solutions in X whose norms are less than q.Section 2 of this article summarizes the reflexive real Banach spaces Lp(x)(Ω), W1,p(x) and some basic theorems of these spaces. Set Set Define the space Then list some basic theorems of these spaces.Theorem 2. Set We haveTheorem 3.Assume mp(x)> n,(?)x∈Ωthere are a continuous compact embedding theorem Wm,p(x)(Ω)→Lq(x)(Ω), where q(x)∈C+(Ω).Section 3,4,5 of this article mainly summarizes the existence and multiplicity of the equation when f(x) satisfy different conditions. Make the appropriate assumptions on f(x,u),p(x),q(x). for example or set etc. Making use of the above theorems, we can get our main results.Theorem 4. Assume the f(x) satisfy the above conditions. Then there exist an open interval A (?) (0,∞) and a positive real number p> 0 such that eachλ∈e∧, the above equation has at least three solutions whose norms are less than p.