Existence Of Positive Solutions For Kirchhoff Equations With Critical Exponents In Four Dimensional Spaces

In this theise, we study the following Kirchhoff type problems where using the Nehari manifold, the concentration compactness principle and the Ekeland variational principle in N = 4.Firstly, we study the Kirchhoff type problems with Critical exponents where f~ C IR4 is a smooth bounded domain, 0 C f~, 0 < q < 1 and A is a positive parameter, we obtain the existence of positive solution. By the methods of Nehari manifold, we can obtain the following result: that for any 0 < A < A,, problem (K1) has at least a positive solutionTheorem 1. Suppose a > O, 0 < b < S--7, then there ezists A, > O, suchSecondly, we study Kirchhoff type problems with Critical exponents and singularity where f~ is a smooth bounded domain in IR4, a, b > 0, 0 C f~ and A > 0 is a real parameter, 1 < q < 2 and 0 </3 < 2Theorem 2. Suppose a > O, 0 < b < ~-~, 1 < q < 2, 0 < /3 < 2, then there ezists A, > O, such that for any 0 < A < A,, problem (K2) has at least a positive solution, when 3 - q <_ /3 < 2, then there ezists 0 < A,-’-~ < ~’,, such that for any 0 < A < A**, problem (K2) has an another positive solution...