Existence And Multiplicity Of Solutions For Critical Equations With Variable Exponents

In recent years,more and more scholars pay attention to the model of partial differential equations with variable exponents which has been widely used in real life.In this paper,we are concerned with critical index equations with variable exponents.By applying to variational methods and the mountain pass theorem,we obtain existence and multiplicity of solutions for critical equations with variable exponents.In chapter 1,we mainly give a general introduction of the background and research status at home and abroad.Then preliminary knowledge and some theorems are introduced.In the last,we give the primary structure of the paper.In chapter 2,we are concerned with a class of fractional equations with critical reaction and variable exponents.We get the existence of non-trivial solutions by using variable exponents,the concentration-compactness principle and the Moser iteration method.Then we generalize the results of the relevant papers.In chapter 3,we consider a class of critical Choquard-Kirchhoff type equations with variable exponents.In non-degenerate and degenerate cases,we prove existence of solutions for equations by the Hardy-Littlewood-Sobolev inequality for variable exponents,the mountain pass theorem and the concentrationcompactness principle.Then in combination with Krasnoselskii genus,we get multiplicity of solutions for equations.