| Fractional Sobolev spaces have been well known since the beginning of the last century,especially in the framework of harmonic analysis.At the same time,the equations with fractional operator have become a very interesting research field in nonlinear analysis.Fractional Sobolev spaces and fractional operators have attracted extensive attention in recent years,both in purely mathematical research and in practical applications.From the viewpoint of practical applications,they have been widely used in various fields.With the continuous emergence of some nonlinear problems in natural science and engineering technology,the limitation of the constant exponent Sobolev spaces and its corresponding operators is gradually revealed.The appearance of variable exponent Lebesgue spaces and variable exponent Sobolev spaces can be described as some nonlinear problems well.For example,the characterization of "pointwise dissimilarity" phenomenon in physics,especially the applications of variable exponent function space and its operator in electrorheology,have made the research on variable exponent problems further developed.In this thesis,we generalize the fractional Sobolev spaces with constant exponent to variable exponent form by using the L∞spaces and the theory of variable exponent Lebesgue spaces,and we study their basic properties and applications in partial differential equations.The main research work of this paper is as follows:1.A new class of variable exponent fractional Sobolev spaces Ws(·),p(·)have been introduced by the extention of constant exponents s and p in the fractional Sobolev spaces Ws,p to the functional form,simultaneously.Firstly,the variable exponent spaces Ws(·),∞have been defined by L∞spaces,based on which the pseudomodular φ is introduced.Secondly,by applying the methods of the theory about variable exponent Lebesgue spaces,the variable exponent fractional Sobolev spaces Ws(·),p(·)have been constructed,following which the properties,named completeness and so on,were studied.2.The compact embedding theorem have been constructed for the variable exponent fractional Sobolev spaces Ws(·),p(·)on the bounded domain.Firstly,in the case of s2(x)≥s1(x),a.e.x ∈Ω,it have been proved that the spaces Ws2(·),p(·)can be embedded continuously into the Ws1(·),p(·).Secondly,by the application of limited coverage theorem into the construction of the finite sets,and the combination of the embedding theorem about the Sobolev spaces,it have been proved that,in the case of s(x)p(x)<n,the spaces Ws(·),p(·)(Ω)can be embedded continuously and compactly into the spaces Lq(·)(Ω),where q(x)<p*(x):=np(x)/n-s(x)p(x)x∈Ω Furthermore,the results in the case of s(x)p(x)=n have been followed.In the above researches,the variable exponents p(·)and q(·)were limited in the case thatp(·)∈[1,∞)and q(·)∈[1,+∞),simultaneously.3.The extension theorem have been obtained about the variable exponent fractional Sobolev spaces Ws(·),p(·).By application of the limited coverage theorem,the unit decomposition theorem and the above compact embedding theorem,the functions in the spaces Ws(·),p(·)(Ω)have been extened to those in the space Ws(·),p(·)(Rn),which were followed by the estimation in norm with respect to the space Ws(·),p(·)(Rn).4.As an application,the Dirichlet boundary value problems in two classes of s(·)p(·)-Laplace equations have been studied.Firstly,based on the critical point theorem with coercive functional,the existence and uniqueness of weak solutions for a class of s(·)p(·)-Laplace equation with nonlinear terms;Secondly,by the application of the compact embedding theorem obtained in this for the variable exponent fractional Sobolev spaces Ws(·),p(·),the Dirichlet boundary value problem of s(·)-p(·)-Laplace equation with the potential function terms have been studied,and the multiplicity problem of weak solutions for such class of equations have been studied based on the mountain pass theorem and the Ekeland variational priciple. |
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