On Naimen’s Open Problem Of The Critical Kirchhoff Equation

In this thesis,we study the following Kirchhoff type equation:(?)where Ω(?)RN(N≥4)is a bounded domain,2≤q<2*,2*=2N/(N-2)is the critical Sobolev exponent and a,b,λ,μ are positive parameters.By using the variational method,we obtain some existence and nonexistence results of(Pa,b,λ,μ)for all N≥ 4 with some further conditions on the parameters a,b,λ,μ,which partially improves some known results in the literature.Furthermore,our result for N=4 and q>2,together with[26]and[27],gives an almost positive answer to Naimen’s open problem.In chapter 1,we introduce the background of Kirchhoff type equations,the progress of Naimen’s open problem and our main work with the corresponding results,respectively.In chapter 2,we introduce some commonly used concepts and lemmas needed in this thesis.In chapter 3,we make some observations on the Nehari manifold by the fibering maps and give a complete description of the(PS)sequences according to the global compactness result in[43],thus giving the existence results of P(a,b,λ,μ)for N=4,q=2 and N=4,2<q<4,respectively.In chapter 4,based on the proof in chapter 3,by finding out a special bounded(PS)sequence of the corresponding functionals and analyzing the compactness of this(PS)sequence carefully,we give some existence and nonexistence results to(Pa,b,λ,μ)for N≥5.