The Applications Of Variable Exponent SOBOLEV Space In Elliptic Differential Equations

With the emergence of nonlinear problems in nature science and engineering,Sobolev spaces demonstrate their limitations in applications. For example, the study ona class of nonlinear problems with variable exponential growth. The nonlinear problemwith variable exponential growth is a new research field. In the studies of this kinds ofnonlinear problems, variable exponent Lebesgue spaces and Sobolev spaces play an im-portant role.In this paper, based on the theory of variable exponent Sobolev space W1,p(x)(?), westudy a class of p(x)-Laplace elliptic equations (systems) and hemivariational inequalitieswith variational structures, where ? ? RN.Note that the exponent p(x) is a function, p(x)-Laplace operator possesses morecomplicated nonlinearities than p-Laplace operator. For example, p(x)-Laplace operatoris inhomogeneous. Thus, some techniques used in the constant exponent case cannot becarried out for the variable exponent case. In this paper, permitting more ?exible require-ments on the growth condition, we first discuss the properties of energy functionals. Then,we study solutions for this kinds of nonlinear problems with p(x)-Laplace via variationalmethod.The main contents are as follows:1. We discuss the weak solutions for a class of p(x)-Laplace equations with subcriti-cal growth. First, we obtain a global minimizer u0∈W1,p(x)(RN) of energy functionalφassociated with the p(x)-Laplace equation, which is a nontrivial critical point ofφ. More-over, we get a nontrivial weak solution for the equation in RN. Next, using the symmetricmountain pass theorem, we get a sequence of critical points {un} ? W1,p(x)(RN) ofφ,which are associated with a sequence of positive energies going toward infinity. Further-more, we obtain the multiplicity of weak solutions for the equation in RN. Finally, we getthe existence of branches of weak solutions for the equation on a bounded domain ? viathe method of sub-supersolution.2. We discuss the weak solutions for a class of p(x)-Laplace systems with subcriticalgrowth. Based on the theory of critical points for strongly indefinite functionals, we get asequence of critical points {(un, vn)} ? W01 ,p(x)(?)×W01 ,p(x)(?) for energy functional I, which are associated with a sequence of positive energies going toward infinity. Moreover,we obtain the multiplicity of weak solutions for the Dirichlet problem of the system on abounded domain ?.3. We discuss the weak solutions for a class of p(x)-Laplace equations involv-ing critical exponent. First, we establish a principle of concentration compactness inW1,p(x)(RN), which generalizes the principle of concentration compactness in Sobolevspace. Then, combining with the symmetric mountain pass theorem, we obtain a sequenceof radially symmetric critical points {un} ? W1,p(x)(RN) for functionalφ, which are as-sociated with a sequence of positive energies going toward infinity. Moreover, we get asequence of weak solutions for the equation in RN.4. We discuss the solutions for a class of hemivariational inequalities with p(x)-Laplace. Applying the non-smooth critical point theory for locally Lipschitz functionals,we study the critical points of locally Lipschitz functionalφ. Then, there exists at leasta nontrivial solution u0∈W01 ,p(x)(?) for the inequalities corresponding to the cases ofbounded and unbounded domain ? ? RN, respectively.Our results are generalization of the corresponding ones with p-Laplace. In addi-tional, our theoretical analyses also re?ect some differences between the nonlinear prob-lems with variable exponent growth and the constant exponent case.