Existence Of Positive Solutions And Sign-changing Solutions For A Class Of Elliptic Systems Involving Multiple Critical Exponents

In this paper, a kind of elliptic systems is studied, which involves critical Hardy-Sobolev exponents, mixed critical terms, subcritical terms and linear terms. The operator (?)is different from those in the past related literatures. The operator is dependent on the positive constant a. The simultaneous appearance of linear terms and subcritical terms brings a lot of troubles. As the existence of critical Hardy-Sobolev exponents and mixed critical terms, the corespponding energy function lose the compactness. So firstly we should establish the local P-S condition.The background and meaning of this elliptic systems is presented in the introduction. We mainly describe from two aspects: the development of critical point theory and the relevant research literature.When q1 = q2 = 2, this kind of system has provided a math model for some physical phenomena related to the equilibrium of anisotropic media. It is significant.In chapter 2, we need to note the analysis of the critical terms in chapter 2. Owing to the critical terms and the mixed critical terms, the P-S sequence is not convergent for all the values of c, only for some of c. Inspired by the previous literatures, by the extremal functions of the best constant, we limit the scope of the value of c, thus we establish the local P-S conditions. This chapter uses variational inequalities and concentration compactness principle.In chapter 3, the positive solution is found by the traditional method. At first, calculate the estimated style of extremal functions. At second, build the mountain geometry of the corresponding energy function. At last, verify the existence of its positive solutions.In chapter 4, by Moser iterative and appropriate analyzed skills, through a lot of calculation, we establish the a symptotic properties of i ts solutions a t t he o rigin, which h elps a l ot i n t he f oregoing p art. I n t his p art, we m ianly a pply m aximum principle, Young i nequalities and H?lder inequalities.Due t o t he f ront s tudy, t he f oundation o f i ts s ign-changing s olution i s f inished successfully. We f ollow t he s ame l ine i n t he l iteratures [ 2-4]. Deformations lemma, Miranda theorem and a basic inequality are used in the proof.