Existence Of Solutions For Kirchhoff-Schrodinger-Poisson System With Two Types Of Nonlocal Terms

With the progress of the society,mutual penetration between mathematics and other disciplines,promote each other,many mathematical equations are the models abstracted from physical,biological,etc.such as:Kirchhoff systems,Schrodinger systems,etc.Kirch-hoff system is putted forward by Kirchhoff’s,which is derived from the elastic string lateral vibration;the Schro dinger-Poisson system is to describe the nonlinear Schrodinger equa-tion with a standing wave model of electrostatic field interaction.These two systems have been studied by many scholars and under the different assumptions on V and f,they have obtained the ground state solution,positive solutions,multiple solutions and sign-changing solutions,But the study of Kirchhoff-Schrodinger-Poisson system is relatively few.So in this paper we will study the existence of solutions for Kirchhoff-Schrodinger-Poisson system under different assumptions.The first chapter,we will study the Kirchhoff-Schrodinger-Poisson equation:where 3 ≤ N ≤ 5.The main aim of this paper is to study the existence of sign-changing of(1.1.1)when the potentials V1 and K decay to zero as |x|→∞.Precisely,we suppose that(V)V1:RN→ R is a smooth function and there exist a,c>0 and τ∈(0,2)such that and V2∈L∞(RN)U L(6-N)N2N(RN)is nonnegative;(K)K:RN→ R is a smooth function and there exist ξ>τ,d>0 such that Related to the function f,we assume that f∈ C(R,R)and satisfies the following hypotheses:(f3)there exists a θ ∈[0,1)such that 1/|t|(K(x)f(t)-θV(x)t)is nondecreasing on(-∞,0)and(0,∞)respectively;(f4]|f(t)|≤C(|t|+|t|p)for some 3<p<5.We prove that the system has a sign-changing solution via a constraint variational method combining with Brouwer’s degree theory.Firstly,we define the corresponding en-ergy functional and manifold M of the system.Secondly,we should prove M is nonempty.Thirdly,the minimizer of the energy functional on M will be found.Finally,we will prove that the system have a least energy sign-changing via a constraint variational method com-bining with Brouwer’s degree theory.In this chapter,we will weak the assumption that f is C1 to that of f being only continuous.We unify the conditions θ＝ and θ>0 and generalize to the problem(1.1.1),then we construct a new homotopy operator to prove the conclusion.The second chapter,we studied the following system:Setting F(x,t)= ∫Ot f(x,s)ds.we introduce the following hypotheses on f:(f1)There exists an open bounded domain Ω such that(?),uniformly for x ∈Ω;(f2)There exists p∈(1,3),q∈(2,4),b ∈ B and a continuous function f which is-τi-periodic in xi with τi>0,i= 1,2,3,such that(a)|f(x,t)-f(x,t)|≤ b(x)(|t| + |t|p-1),(x,t)∈ R3 × R;(b)thr function t→f(x,t)/|t|qt is non-increasing in(-∞,0)and non-decreasing in(0,∞);(c)F(x,t)≤ F(x,t)=∫0t f(x,s)ds,(x,t)∈ R3 × R;(f3)(?),uniformly for x ∈R3,and there exist 3<r<5,h∈L3/2 ∩L6/5-r∩such that for all x∈ RN and t∈R|f(x,t)| ≤ h(x)(|t| + |t|r).V satisfy the following hypotheses:(V)there exists a continuous and periodic function V,τi-periodic in xi with τi>0,i=1,2,3,such that V-V ∈ B and 0<V0≤V(x)≤ V(x)for all x ∈ R3.Where B = {b ∈ C(R3)∩ L∞(R3):m({x ∈R3:|b(x)| ≥ε})<∞},m denote the Lebesgue measure.Because of the nonlinearity have no monotonicity conditions and the nonlocal∫R3 |▽u|2 makes proved relatively difficult.We will use the mountain pass lemma to prove that the existence of the system.