Positive Solutions For Two Classes Of P-Kirchhoff Type Equations

In this paper,the existence of positive solutions for two classes of p-Kirchhoff type equations is investigated by using variational methods.Firstly,the existence of positive solutions for p-Kirchhoff type equation with critical exponent is considered.Secondly,the existence,non-existence and exponential decay of solutions for p-Kirchhoff type equation with a parameter is studied.The main theoretical basic is mountain pass lemma,concentration-compactness principle,implicit function theorem,optimization theorem and comparison principle.In chapter 3,the following p-Kirchhoff type equation with critical exponentTheorem 1.1 If one of the following conditions is satisfied,In chapter 4,the following p-Kirchhoff type equation with a parameter is considered,where 1<P<N,b>0 is a parameter,V,f satisfy the following hypotheses:The main results is as follows.Theorem 1.2 Suppose that?V₁?,?V₂?,?f₁?and?f₂?satisfies,then?i?there exists b*>0 such that the equation?P_b?has a positive solution for b ??0,b*?which possesses exponential decay at infinity,i.e.there exists C,R.?>0 such that u?x?? Ce-?|x|,|x|>R.?ii?for 2p<p*there exists b>0 such that the equation?P_b?has no nontrivial solution for b ? b.The structure of this paper is as follows.In chapter 1,we introduce the research development of Kirchhoff type equations and give the main results of this paper.In chapter 2,we give some necessary preliminaries of the variational methods of this paper.In chapter 3,we give the variational structure of p-Kirchhoff type equation with critical exponent and consider the existence of positive solutions for the p-Kirchhoff type equation by using variational methods and concentration-compactness principle.In chapter 4,we give the variational structure of p-Kirchhoff equation with a pa-rameter,and by using variational methods and comparison principle,we study the existence and exponential decay of positive solutions to p-Kirchhoff type equation for b sufficiently small,and the non-existence of nontrivial solutions to the equation for b sufficiently large and 2p<p*.