Gagliardo-Nirenberg Inequality And Critical Neumann Problem With Hardy Potentials

In this dissertation, we discuss the best constant problems of Sobolev inequalities, Gagliardo-Nirenberg inequalities and related inequalities on Riemannian manifold and the existence of solutions for semilincar elliptic equations with critical exponent. It comprises five chapters.In Chapter 1, we give a survey on the background and recent development of the best constant problems of Sobolev inequalities, Gagliardo-Nirenberg inequalities and so on, and then recall the recent research of Hardy inequality and elliptic equations with critical exponent. Finally, we simply list our work.In Chapter 2, we discuss the first best constant problem of improved Gagliardo-Nirenberg inequalities and get that the first constant of this inequality is attained by contradiction argument.In Chapter 3, we recall the related knowledge of vector valued maps and the recent results for the best constant problems of Sobolev inequalities under vector valued maps, and then prove that the second best constant A0(g) is continuous dependent of the metric g in C2-topology.Chapter 4 deals with the following semilinear elliptic equations with Hardy potential and critical Hardy-Sobolev exponent:We show the existence of positive solutions by Mountain-Pass Lemma and Concentration-Compactness principle.In Chapter 5, we prove a Hardy inequality in Warped product manifold: and show its geometric applications in conformal geometry.