Existence Results for Brezis-Nirenberg Problems with Hardy Potential and Singular Coefficients.pdf

1、In this dissertation, we consider the existence of non-trivial solutions to semi-linear BrezisNirenberg type problems with Hardy potential and singular coefficients. And the existence of multiple positive solutions for s

2、ingular elliptic equations involving a concave term and critical Caffarelli-Kohn-Nirenberg exponent. Some new conclusions are obtained. The main contents of the dissertation are as follows:
　　 Chapter one summarize s

3、tatement of the following problems: the existence of non-trivial solutions to semi-linear Brezis-Nirenberg type problems with Hardy potential and singular coefficients. And the existence of multiple positive solutions fo

4、r singular elliptic equations involving a concave term and critical Caffarelli-Kohn-Nirenberg exponent. Moreover summarize of the main results.
　　 In chapter two we shall study the corresponding eigenvalue problem, a

5、nd get some basic properties of eigenvalues and asymptotic estimates of the eigenfunctions and approximating eigenfunctions.
　　 In chapter three, we consider the extremal functions of the best embedding constant, and

6、 applying Bliss Lemma([1]) to prove the explicit form of extremal function. And for the use in next chapter derive some estimates of the cut-off functions of extremal functions, which shows the concentration of extremal

7、functions near x = 0.
　　 Chapter four, applying different variational for distinct cases of those parameters in the equation, first we prove that Iλ, μsatisfies the (PS)c condition if the energy level β is under a th

8、reshold. Secondly, we recall some variational principles, and show that Iλ, μsatisfies the geometric conditions of those variational principles in each case, hence get the (PS)β sequence for some minimax values β. Thirdl

9、y, by a fine balance between the blow-up of eigenfunctions and concentration of extremal functions near x = 0, we try to show that minimax values β are under the threshold, and thus obtain two existence results of non-tr

10、ivial solutions to semi-linear Brezis-Nirenberg type problems in resonant case and non-resonant case.
　　 In chapter five, we shall apply Ekeland Varitional Principle to obtain the first nonnegative solution of (ρλ,μ)