Existence And Multiplicity Of Nontrivial Solutions To Two Classes Of Kirchhoff Equations

The Kirchhoff type equation is a typical non-local elliptic differential equation which is used to describe the changes in length of the string produced by transverse vibrations.And it also be widely used in the fields of cosmic physics,non-Newtonian mechanics,population dynamics and so on.Due to the presence of Kirchhoff term,equations are viewed as being non-local.This observation brings mathematical challenges to the analysis,and at the same time,makes the study of such a problem particularly interesting.This dissertation uses the variational method to study the existence and multiplicity of nontrivial solutions to Kirchhoff type equations and system.In the first chapter,we mainly introduce the research background of Kirchhoff-type equations and the development status at home and abroad.In addition,we also give a brief introduction to the research results of this article and give the necessary knowledge for the paper.The second chapter of this paper mainly studies the existence of ground state solutions and multiplicity of nontrivial solutions for the following Kirchhoff type equations with sub-critical growth nonlinear-terms:where a>0,b? 0 are positive parameters,and 1<p<5.If we use critical point theory to study the existence of solutions to the above equation,it is quite difficult to prove the boundedness of the Palais-Smale(PS for short)sequence,since the well-known Ambrestti-Rabinowitz(AR for short)condition does not hold.In order to overcome this difficulty,we employ a perturbation approach to study a modified equation with perturbation coefficients,and then make use of the well-known mountain pass lemma to prove the existence of positive solutions of the modified equation.Finally,we use the Pohozaev identity to prove the existence of positive ground state solutions of the original equation by letting the perturbation trend to zero.Moreover,we use the perturbation method combined with symmetrical mountain pass lemma and Pohozaev-type identity to prove the existenc of infinitely many non-trivial solutions to the Kirchhoff equation.The third chapter mainly studies the existence of positive solutions and infinitely many nontrivial solutions to the following Schrodinger-Kirchhoff system where a>0,b?0,?1,?1(i=1,2)are positive parameters and q ?(2,3)Notice that the standard Sobolev space embedding in the whole space is not compact.We use the Nehari manifold method toegther with Schwarz symmetric rearrangement technique to prove that the above system has at least one positive ground state solution u1*(i=1,2)where are radially symmetric.Moreover,we use the symmetric mountain path lemma to prove that the above system has infinitely many high-energy solutions.