The Existence Of Solutions For A Class Of Variable Exponent Partial Differential Equations With Critical Frequency

Variable exponent spaces were thoroughly studied by Kovacik and Rakosnik firstly,which have been used to the study of non-standard variational problems and PDEs in the previous decades,such as in the electrorheological fluids,thermorheological fluids,image processing and so on.For the nonlinear problems with variable growth,there have been a great deal of research activities on the existence,multiplicity,uniqueness and regularity of solutions.Variable exponent Lebesgue spaces and Sobolev spaces play an important role in the study of this kinds of nonlinear problems.In this thesis,based on the theory of Sobolev space W^1,p?x????,we study a class of p?x?-Laplace equations with critical frequency -div(|?u|^p?x?-2?u)+ V?x?|u|^p?x?-2u= f?x,u?,X?R^N where 1<p-?p?x??p+<N.What is different from the previous results is that the potential function V of this kind of equation can satisfy that minV?x?=0 and the right-hand side function of the equation is subcritical growth,which is a relatively general condition.The details are as follows:Firstly,because the embedding of W^1,p?x??R^N?to L^q?x??R^N?is only continuous but not compact,the Palais-Smale condition is difficult to verify.We discuss the properties of energy functional under more flexible requirements on the growth condition.Although,for any c,it is not possible to prove the functional satisfies the?PS?c condition,we can give c a range,which is c,such that the functional satisfies the?PS?c condition over the range of[0,c].Secondly,we prove the functional has the geometric properties of Mountain Pass lemma.Finally,by using the minimax principle,we obtain the existence of nontrivial weak solutions.Moreover,if f?x,u?is odd with respect to u,for any m?N we derive the existence of m pairs of nontrivial solutions.