Some Moser-Trudinger Inequalities And Their Extremal Problems

The Moser-Trudinger inequality is a very important class of inequalities.It is widely used in the problem of the existence of the solution of the equation and the geometric analysis.In the past twenty or thirty years,the Moser-Trudinger inequality and its corresponding extremal problems have been widely studied,and fruitful results have been achieved.But with the development of the blow up anal-ysis of the Liouville equation with singular points,the Moser-Trudinger inequality with singular points and the corresponding extremal problem has become one of the hot spots of the geometric analysis and the elliptic partial differential equation A lot of mathematicians recently work on the Finsler-Laplacian(anisotropic Lapla-cian)equation and work on the Finsler-Liouville equation,and the fruitful results have been achieved.It provides a theoretical basis and research motivation for the study of anisotropic Moser-Trudinger inequalities.In view of the above basis.we study the Moser-Trudinger inequality with the anisotropic norm and a class of Moser-Trudinger inequality with singular points and the related extremal problem.and we obtain the corresponding results.The specific content of this article can be summarized as follows.In the first chapter,the research background of this paper and the research progress of Moser-Trudinger inequality are briefly introduced,and the main con tents of this paper are also introducedThe second chapter,we introduce the related preparatory knowledge in this paper,including the properties of anisotropic function,F(x),and the regularity theory of anisotropic-Laplacian equationIn the third chapter,we mainly study the anisotropic Moser-Trudinger in-equality and extremal problem on Wullf balls in R2.We have proved by the blow up analysis,the level-set method and the convex symmetry rearrangement method We obtain the results on the anisotropic Moser-Trudinger inequality and the exis-tence of the extremal value function.In the fourth chapter,we mainly study the existence of the anisotropic Moser-Trudinger inequality and its extremal function in the bounded domain of Rn.Be-cause of the generality of the region,the rearrangement method in the third chapter is no longer applicable.To this end,we have established an anisotropic P.Lions lemma,which provides a theoretical basis for blowing up analysis,and then we has completed the proof of the theorem.In the fifth chapter,we mainly study the existence of anisotropic Moser-Trudinger inequalities unbounded regions and the existence of their extremal func-tions in the whole space Rn.By using the existence results in the fourth chapter,we obtain a maximizing sequence of functions for consided problem,and then by means of asymptotic expansion of anisotropic Green function and blowing up anal-ysis techniques,we prove the Moser-Trudinger inequality and the existence of the extremal function on Rn.In the sixth chapter,the Moser-Trudinger inequality with several singular points and the remaining terms in a two-dimensional region and the relative ex-tremum problem is studied.Using the singularity positioning technique,we prove the Moser-Trudinger inequality with several singularities and remainders on a two-dimensional bounded domain.We also obtain the existence result of the extremal function.