Best Constants And Extremal Functions For Sobolev Inequalities

Sobolev inequalities also called Sobolev imbedding theorems, are very popular among writers in partial differential equations or in the calculus of variations. In this paper, we consider the following Sobolev imbedding inequalities, Hardy-Sobolev inqualities and Caffarelli-Nirenberg-Sobolev inequalities:where 0＜α＜n, q =(?).where 1＜p＜n, 0≤s＜p and p≤q = (?).where -∞＜a＜(?), a≤b≤a + 1, p= (?).Remarks: We will precisely define (?) in section 2.Using the known results, we shall summarize the existence and the propertiesof the extremal functions for the above inequalities. We shall summarize the radial symmetric of the extremal functions by the spherically symmetric rearrangement(also called Schwarz symmetric rearrangement), and summarize the precise expressions of the extremal functions and the best constants by the polar coordinatetransformation (or by the Fourier transformation). Moreover we summerize the solutions of the corresponding partial differential equations. When we discuss the existence of solutions for the partial differential equations, we usually use the Mountain Pass Theorem. In the end of this paper, we will give some examples for the application of the extremal functions. The main results in this paper mainly come from [1], [9], [6], [8], [15], [18], [5].