Existence And Non-existence Of Extremum Functions For Two Class Of Variational Problems With Weighted Constraints

In this paper,we study the existence and nonexistence of extremum functions for a class of variational problems with weighted constraints,and discuss the Trudinger-Moser inequality involving the anisotropic norm with logarithmic weights and the existence of the extremum functions.In the first part,we study the existence and nonexistence of extremum functions for weighted variational problems associated with Caffarelli-Kohn-Nirenberg type inequalities both in the subcritical case and critical case.Firstly,we establish the one-dimensional functions expression of the variational problems,and give the relationship between the attainability of the supremum in variational problems and the attainability of the supremum of some one-dimensional functions.Secondly,we obtain the existence and nonexistence of extremum functions for the variational problems by studying the attainability and unattainability of the supremum of one-dimensional functions.Furthermore,if the extremum functions exist,we can also deduce the concrete form of it.In the second part,we study Trudinger-Moser inequality with logarithmic weights and the existence of extremum functions in dimension two.Firstly,we establish a critical radial lemma,then use it and a test function to get the subcritical Trudinger-Moser inequality involving the anisotropic norm with logarithmic weights and its sharp constant.Secondly,we prove the critical inequality by using the famous Leckband’s inequality.In particular,we observe a sharp Trudinger-Moser growth of double exponential type when the index of the weight function is 1.At last,we give the proof of a concentration-compactness principle of Lions type and the continuity of the upper bound of functional energy on exponents.Moreover,when the index of the weight function is sufficiently close to 0,we get the existence of extremum function for Trudinger-Moser inequality.