Research On Some Types Of Trudinger-Moser Inequalities

In this thesis,we study the several Trudinger-Moser inequatities problems:concentration-compactness principle for Trudinger-Moser inequality under Lorentz norm on the whole space,critical and subcritical anisotropic Trudinger-Moser inequalities on the entire euclidean spaces,concentration-compactness principle associated with Adams inequality in Lorentz-Sobolev space.At last we consider the effect of lower order perturbation on the attainability of the supremum of the Trudinger-Moser inequality.In the first part,we consider the concentration-compactness principle for Trudinger-Moser-Lorentz type inequalities on the Lorentz space.According to the characterization of Lorentz space,we consider two cases q∈(1,N]and q>N.We use the properties of rearrangement function to turn the N-dimension integral into 1-dimension integral,then use the method of level sets of truncation to prove the boundedness of the integral.When q>N,we need to use the equivalence of Lorentz norm.Additionally,we establish the test function sequence to verify the sharpness of the conclusion.It is worth mentioning that the method of level sets of truncation is simpler and more straightforward than the method with symmetrically rearrangement.Using this argument,the concentration compactness-principle on the compact Riemannian manifolds and Heisenberg group can also be established.In the second part,we first study the subcritical Trudinger-Moser inequality under anisotropic norm on the whole space and establish the asymptotic behavior of the supremum.Then,by using above result,we study the critical Trudinger-Moser inequality under anisotropic norm and verify the sharpness of the result.For more,we provide a precise relationship between the supremums for the critical and subcritical inequality under anisotropic norm on the whole space.In the third part,we study the concentration-compactness principle of second order Adams inequality in Lorentz-Sobolev space W₀²L^2,q（（?））.Because the Pólya-Szeg（?）inequality is no longer ture with respect to the second order derivatives,we will use a symmetrization-free argument to overcome the difficulty.Furthermore,we also show the sharpness of the result by constructing a test function sequence.In the fourth part,we study a maximization problem on the Trudinger-Moser inequality.Specificly,we consider the effect of lower order perturbation on the attainability of the inequality on R².Assuming the blow-up does not occur,we prove that the blow-up must occur when the perturbation increases infinitely.The estimation of Lebesgue term and exponential term is studied by using the comparision principle and quantization results,and the contradiction is derived.It shows that the blow-up will occur when the lower order disturbation is large enough,then we can give the explicit forms of threshold nonlinearity involving the existence of Trudinger-Moser inequalities’extremal function.