A priori bounds and existence of positive solutions of an elliptic system of Kirchhoff type in three or four space dimensions

Let be a bounded and smooth domain. Assuming that , , we consider the following elliptic problem of Kirchhoff type Under some assumptions of , we get a priori bounds of the positive solutions to the problem (0.1) when and p, q satisfy , and . By these estimates and the method of continuity, we get the existence for the positive solutions to the problem (0.1). Moreover, the effect of the term on the solution set of the problem (0.1) also can be given in section 2.