Existence And Multiplicity Of Solutions For Two Classes Of Nonlocal Problems

In this paper, we firstly study the existence and multiplicity of nontrivial solutions for a new nonlocal problem with Dirichlet boundary value conditions by using variational method and mountain pass lemma. Secondly, the existence of positive solutions for three kinds of typical Kirchhoff type problems in RN and the nonexistence results of the nontrivial solutions for Kirchhoff type problems associated with them in critical or supercritical conditions wasw investigated by using the well-known results to elliptic problem and the scalings this paper proposed.Firstly, we study the existence and multiplicity of nontrivial solutions for the following nonlocal problem with Dirichlet boundary value conditions Where Ω is a smooth bounded domain in RN, N ≥ 1, a, b> 0, andThe main results is as follows. Theorem 1 The problem (P1) possesses at least a nontrivial weak solution in H01(Ω).Theorem 2 The problem (P1) possesses at least a nontrivial non-negative solution and a nontrivial non-positive solution in H01(Ω).Secondly, we investigate the existence of positive solutions for the following three kinds of typical Kirchhoff type problems and-(α+λ∫RN|▽u|2dx)△u+μu=|u|p-1u,x ∈RN (P4) in RN.Where the dimension N≥2,λ>0 is a parameter,α>0 is a constant,μ>0,and 1<p<2*-1.Meanwhile,we study the nonexistence of nontrivial solutions for Kirchhoff type problems and Where N≥3,p≥2*-1,α,b,μ>0 are constants.Let U be the only positive radial solution of -△u+u=|u|p-1u,x ∈RN.The main results is as follows.Theorem 3(1)Suppose N=2,3 and 1<p<3,or N≥4 and 1<p<2*-1.Letting then the problem(P2)has two positive solutions if λ ∈(0,λ0),one positive solution if λ=λ0 and no nontrivial solution if λ ∈(λ0,∞).(2)Suppose N=2,3 and p=3.Letting then the problem(P2)has one positive solution if λ ∈(0,λ1)and no nontrivial solution if λ ∈[λ1,∞).(3)Suppose N=2,3 and 3<p<2*-1.Then the problem(P2)has one positive solution.Theorem 4 (1) Suppose N= 2. Letting then the problem (P3) has one positive solution if λ ∈ (0, λ0) and no nontrivial solution if λ ∈[λ0,∞).(2) Suppose N ≥ 3. Letting where then the problem (P3) has two positive solutions if λ ∈ (0, λ1), one positive solution if λ= λ1 and no nontrivial solution if λ (λ1,∞).Theorem 5 (1) Suppose N= 2,3. Then the problem (P4) has one positive solution for any λ> 0.(2) Suppose N= 4. Letting then the problem (P4) has one positive solution if λ ∈ (0, λ0) and nonontrivial solution if λ ∈[λ0,∞).(3) Suppose N ≥ 5. Letting then the problem (P4) has two positive solutions if λ ∈ (0, λ1), one positive solution if λ= λ1 and no nontrivial solution if λ ∈ (λ1,∞).Theorem 6 If N ≥ 3,p ≥ 2* - 1 and α, b> 0 are constants. Then 0 is the only solution of problem (P5).Theorem 7 If N ≥ 3, p ≥ 2*-1 and α, b> 0 are constants. Then 0 is the only solution of problem (P6).Theorem 8 If N ≥ 3,p ≥ 2*-1 and α,b,μ> 0 are constants. Then 0 is the only solution of problem (P7).The structure of this paper is as follows.In the first chapter, we introduce the background and research advance in recent years of nonlocal problems and Kirchhoff type problems.In the second chapter, some necessary preliminaries of the variational method are stated.In the third chapter, we firstly give the variational framework of problem (Pi). Secondly, we give the proofs of the existence and multiplicity of nontrivial solutions for problem (Pi) by using mountain pass lemma.In the fourth chapter, three kinds of Kirchhoff type problems has changed into three kinds of systems of problems each including a differential equation and an integral equation by using the scalings this paper proposed respectively. And then the existence of nontrivial solutions for problems (P2)-(P4) are obtained by using the well-known results to ellipse prob-lem. Furthermore, we give the proofs of the nonexistence of nontrivial solutions for (P5)-(P7) in the same way.In the fifth chapter, the main work of whole paper is summarized and the next research program is discussed.