The Existence Of Nontrivial Solutions For P-Kirchhoff Type Equations In R~N

The existence of nontrivial solutions for p-Kirchhoff type equations with critical nonlinearities in RN and the existence of two nontrivial solutions for p-Schrodinger-Kirchhoff type equations in RN are investigated by using variational methods.The main theoretical basis are Mountain pass lemma and concentrated compactness principle.Firstly,The existence of nontrivial solutions for p-KirchhofF type equation is investigated,where a,b,m>0 are positive constants,1<p<N,p*=Np/N-p,?>0 is parameter,p-Laplacian operator ?pu:=div(|?u|p-2?u).The main result is as follows.Theorem 1.Assume p2<p*,then when one of the following conditions hold,the equation(P1)has at least one nontrivial solution.Secondly,the existence of two nontrivial solutions for p-Schrodinger-Kirchhoff type equation is investigated,where a,b>0 are constants,1<q<2p<<p*.Functions V and g satisfy the following assumptions:The main results is as follows.Theorem2.When conditions(V)and(g)are satisfied,the equation(P2)has at least two nontrivial solutions,one with positive energy,the other with negative energy.The structure of this paper is as follows.In the first chapter,the basic theory of variational methods are introduced,in recent years the authors have studied of Kirchhoff type equation by variational method and the new progress of Kirchhoff type equation are also introduced.Then the main research contents of this paper are stated as well as the conclusions obtained.In the second chapter,the preliminary knowledge required to prove the existence of nontrivial solution for equation(P1)are stated,and the specific process of prove the main result is given.In the third chapter,the preliminary knowledge required to prove the existence of two nontrivial solutions for equation(P2)are stated,and the specific process of prove the main result is given.