Research On The Several Trudinger-Moser And Adams Inequalities

Trudinger-Moser and Adams inequalities play a wide role in the problems of the existence of solutions to partial differential equations,energy estimates and nonlinear analysis.In this thesis,we study Trudinger-Moser inequalities in weighted Sobolev space and Adams inequalities in Lorentz-Sobolev space.To be specific,the research can be divided into the following three parts:In the first part,we study the singular Trudinger-Moser inequalities on weighted Sobolev Spaces.Firstly,in order to simplify the problem,the Schwartz symmetric rearrangement method is used to convert the original problem to the one dimension problem,then the inequalities in the critical case can be proved by using the abstract Leckband inequality and the asymptotic estimate of the function near the origin obtained through the radial lemma.Finally,an appropriate test functions is constructed to verify the optimality of the inequality.In the second part,we study the singular Adams inequality on Lorentz-Sobolev space.Firstly,the asymptotic estimate of the function near the origin is obtained by using a rearrangement inequality established by A.Alberico,and the inequality in the critical case is proved by the one-dimensional lemma of J.Moser.Finally,we construct suitable test functions to verify the optimality of the inequality.In the third part,we study the sub-critical Adams inequalities in Lorentz-Sobolev space defined on the four-dimensional full space.Firstly,an equivalent form of the Adams inequalities is obtained by using the scaling transformation.Secondly,based on the rearrangement-free argument,the function considered is divide into two parts according to the size of the absolute value.For the part with small absolute value,the bounded of the integral is proved by using the relationship between the integral and the distribution function.For the part with large absolute value,the asymptotic upper bound of the subcritical Adams functional is obtained by through the Admas inequality with Navier boundary condition.Finally,the asymptotic lower bound of the Adams functional is obtained by constructing a suitable test functions.