Existence And Multiplicity Of Positive Solutions For Two Classes Of P-kirchhoff Type Equations

In this paper,the existence and multiplicity of positive solutions for two classes of -Kirchhoff type equations is investigated by using variational methods.Firstly,the existence of positive ground state solutions for -Kirchhoff type equation with critical exponent is considered.Secondly,the existence and multiplicity of positive solutions for -Kirchhoff type equation with sublinear term is studied.The main theoretical basic is mountain pass lemma,Ekeland variational principle,concentration-compactness principle,vanishing lemma,implicit function theorem and optimization theory.In the second chapter,the existence of positive ground state solutions for the following -Kirchhoff type equation with critical exponentis considered,where (6,(7 are positive constants,???satisfies the following hypotheses:?f₁?f is odd and f?s?> 0,s>0;?f4?There exist ?> 0 and q?(p,2p] such that f?s???s^q-1,s>0;?f5?There exists ??(p,2p] such that ??? is strictly increasing in?0,??.The main result is as follows.Theorem 1.1 Assume that?f1?-?f5?satisfy and ? is sufficiently large,then the problem?P₁?has a positive ground state solution.In the third chapter,the existence and multiplicity of positive solutions for the following p-Kirchhoff type equation with sublinear termis considered,where (6,(7 are positive constants, > 0 is a parameter,???R satisfies the following hypothesis:The main results are as follows.Theorem 1.2 Assume that f is a sign-change function and the hypothesis?f?holds,then there exists ?1> 0 such that for every ???0,?1?,the problem?P₂?has a positive solution which has a negative energy.Theorem 1.3 Assume that f is a nonnegative function and the hypothesis?f?holds,then there exists ?2??0,?1?such that for every ? ??0,?2?,the problem?P2?has two positive solutions,one of which has a positive energy and the other has a negative energy.The structure of this paper is as follows.In the first chapter,the research development of Kirchhoff equation in recent years is briefly introduced.Then the research work and main results of this paper are presented.In the second chapter,some preliminaries and the proof of the main lemmas to prove the existence of positive ground state solutions for -Kirchhoff type equation with critical exponent are presented.Then the proof of the main result is given.In the third chapter,some preliminaries to prove the existence and multiplicity of positive solutions for -Kirchhoff type equation with sublinear term is presented.Then the proofs of the main results are given.