The Existence And Concentration Of Bound State Solutions To Kirchhoff-type Problem

In this paper,we will use the Minimax principle and the quantitative deformation principle on manifold to study the following Kirchhoff-type equation where a>0,b?0,N?3,the potential function V and nonlinearity h satisfy appropriate conditions.If there is a nontrivial solution and the L2-norm of the solution is a given positive constant,we call that the solution with this property is a bound state solution.When b=0,the above equation is a classical Schrodinger equation.It has a wide range of physical background,the vibration tending to be balanced,the heat conduction tending to be stable,and the conservative field can be summed up as such equation.When b?0,as a Kirchhoff-type equation,it has a wide range of applications in physics and biology.For example,in the D'Alembert's wave equation u represents displacement,and in biological system u represents population density.This paper consists of five chapters.In the first chapter,we introduce the research background,the research advance and the main conclusions on Kirchhoff-type equation in this paper.In Chapter 2,we will use the Minimax principle on manifold to study the following Kirchhoff-type equation with Hartree nonlinearity#12where a>0,b? 0,N?3,??(max{0,N-4},N)and p ?((N+?)/N,(N+?)/(N-2)).When p ?((N+?)/N,(N+?+4)/N),I satisfies the coercive condition,thus I is bound from below under the constraint condition of Sc:={u ? X:|u|22=c2},that is,for any c>0,ic:=inf Sc I>-?,where X is a Banach space denoted by H1(RN)or Hr1(RN).In addition,when p ?((N+?+4)/N,(N+?)/(N-2)),I is not bounded on Sc,that is,for any c>0,ic=-?.However,when p=(N+?+4)/N,for any c>0,we can not determine ic>-? or ic=-?.We say that p=(N+?+4)/N is the L2-critical exponent,p ?((N+?)/N,(N+?+4)/N)and p ?((N+?+4)/N,(N+?)/(N-2))are called L2-subcritical and L2-supercritical cases,respectively.In this chapter,we systematically consider the existence of constrained state solutions for the above Kirchhoff-type equation.Firstly,we give the sharp existence of minimizers in the case of L2-subcritical.On this basis,we prove that the energy functional I exists the mountain pass structure on the constraint condition Sc.Furthermore,there are bound state solutions.Therefore,the functional I has two critical points on the constraint Sc,which are global minimizer and mountain pass solution.Then,the existence of bound state solution is verified and the concentration of solution is analyzed in the case of L2-critical.Finally,the existence of bound state solution is proved by using global compactness lemma and subadditivity of mountain pass value in the case of L2-supercritical.First,we discuus the sharp existence results of the minimizers about ic.Theorem 2.1.1 Let p ?((N+?)/N,(N+?)/(N-2)).(1)if p?((N+?)/N,(N+?+4)/N),then(?)there exists such that for any c ?(0,c*],ic=0 and c ?(c*,?),ic<0;(?)ic has minimizers if and only if(?)(2)if p=(N+?+4)/N,for any c ?(0,(2-1b)N/[2(?+4-N)]|Qp|2(?+4)(?+4-N),ic=0 and c ?((2-1b)N/[2(?+4-N)]|Qp|2(?+4)/(?+4-N),?),ic=-?.Moreover,for any c>0,ic does not have minimizers.(3)if p ?((N+?+4)/N,(N+?)/(N-2)),then ic=-?,thus for any c>0,ic does not have minimizers.Based on the conclusions of the above theorem,the minimizers of ic are the global minimizers of ic on Sc.According to the Lagrange multiplier method,there exists ??R such that(?,u)?R × Sc is the bound state solution to the equation(2.1.1).Next,we state the existence theorem of bound state solutions in the case of L2-subcritical case.Theorem 2.1.2 Let p ?((N+?)/N,(N+?+4)/N).For any fixed c>0,if uc is the minimizer of ic,then there exists ?c<0,such that(?c,uc)?(-?,0)× Sc is the solution to the equation(2.1.1).In the third chapter,we study Kirchhoff-type equation with potential function#12 where potential function V satisfies(V)V ? Lloc?(RN),infRN V=0 and lim|x|?? V(x)=?.Due to ?RN V(x)u2 becomes ?RN V(t-1x)u2 after the action of T,the subadditivity of the mountain pass value in Lemma 2.2.6 and the mountain pass structure of IV do not hold,so the minimizers of iV(c)and the solutions of mountain pass in the case of L2-subcritical do not exist,where iV(c):=infSc IV,IV is the energy functional related to the above equa-tion,Sc:={u ?HV1(RN):|u|22=c2} and HV1(RN):={u?H1(RN):?RN V(x)u2<?}.However,according to the results of Theorem 2.1.1,firstly,we can prove the existence of the minimizers of the problem(3.1.1)under the condition of(V).In addition,We will use mathematical analysis to give the relationship between the minimizer uc and c(equality(3.3.7)).Then,the concentration of minimizers are analyzed.Firstly,we can prove the sharp existence of the minimizers about iV(c).Theorem 3.1.1 Let p ?((N+a)/N,(N+?+4)/N].Then(?)iV(c)exists minimizers.Moreover,there exists(?c,uc)?R×Sc is the solution to the equation(3.1.1)and IV(uc)=iV(c).In the following,when p ?((N+?)/N,(N+?+4)/N)\{2} and c??,we study concentration phenomenon of the minimizers of iV(c).Theorem 3.1.3 Let p ?((N+?)/N,(N+?+4)/N)\{2}.For any sequence{cn} C(0,?)satisfying cn?oo as n??,and {un}(?)Scn are the minimizer of iV(cn).Then there exists the subsequence of {cn}(still denoted by {cn})and {zn}(?)RN such that for any q ?[2,2*),as n??, in Lq(RN),where(?),D1=(Np-N-a)/N,D2=(N+?+4-Np)/[4(Np-N-?)],zn=rcyn/c and Wp is the nontriv-ial solution to the equation(2.1.4).In Chapter 4,we study the following Kirchhoff-type equations with nonlocal terms#12 where a>0,b?0,N?3,? ?(N-1,N)and p ?((N+?)/N,(N+?+4)/N).Inspired by the fact that the global and local minimizers of Schrodinger-Poisson equation are considered in the reference[89],we consider the minimizers of Schrodinger-Poisson equation with Hartree nonlinearity,so we firstly need to prove the existence of the minimizers.Since the lack of Pohozaev identity about the above equation,we can only prove the existence of global minimizers in the case of L2-subcritical.In addition,when b=0,due to the appearance of nonlocal term ?uu,we will use different methods from those in sections §2.4 and §3.4 to analyze the concentration phenomenon of minimizers.Next,we give the existence of restricted minimizers in the case of L2-subcritical.Theorem 4.1.1 Let p ?((N+?)/N,(N+?+4)/N).Then there exists c*>0 such that for any c>c*,J possesses minimizer uc on the constraint Sc.Moreover,there exists?c?R such that(?c,uc)?R×Sc is the solution to the equation(4.1.1).Similar to the proof of Theorem 4.1.1,when b=0,we prove that the equation(4.1.1)degenerates into the Schrodinger-Poisson equation.We can obtain the existence of the following bound state solutions.Theorem 4.1.2 Let b=0 and p ?((N+?))/N,(N+?+2)/N).Then there exists c*>0 such that for any c>c*,J has the minimizer uc on constraint Sc.Moreover,there exists ?c ?R such that(?c,uc)?R×Sc is the solution to the equation(4.1.1).Finally,we state the concentration properties of the bound state solutions to the equation(4.1.1).Theorem 4.1.4 Let b=0 and p ?((N+?)/N,(N+?+2)/N)\{(N+?+3)/(N+1)}.For any sequence {cn} that satisfies one of the following conditions(i)when p ?((N+?)/N,(N+?+3)/(N+1)),limn?? cn=0;(ii)when p ?((N+?+3)/(N+1),(N+?+2)/N),limn?? cn=?,there exists the subsequence of {cn}(still denoted by {cn})and {zn}(?)RN such that the minimizers {un}(?)Scn satisfying converge to Wp in Lq(RN)for any q ?[2,2*),where l=1/(N+?+2-Np),zn=t1yn,and Wp is the nontrivial solution to the equation(2.1.4).Finally,in Chapter 5,we study the following Kirchhoff-type equations with general nonlinearity where a>0,b?0,nonlinearity f?C(R)satisfy the appropriate conditions.Since there is no concrete expression of the nonlinear term,it is difficult to prove the existence and global compactness lemma of the minimizers.We obtain the existence of the bound state solutions by finding the local minimizers.The main conclusion in Chapter 5 is the following theorem.Theorem 5.1.2 Let(f1)-(f4)hold.For any c>0,there exists ?c?R such that(?c,uc)?(-?,0)× Sc is the solutions to the equation(5.1.1).