Existence And Concentration Of Solutions To Kirchhoff-type System With Hartree-type Nonlinearity

Nonlinear partial differential equation usually arises in the natural science and engineer-ing areas and has a wide range of background. Because they can well explain the important natural phenomenon and solve the nonlinear problems effectively, a large number of science researchers have paid attention to the problems for a long time. In this paper, we discuss the existence and concentration of positive ground state solutions and sign-changing solutions to a class of Kirchhoff-type systems with Hartree-type nonlinearity. First, by the variational method, we prove that the system has positive ground state solutions when the coefficient of the potential function is larger than a threshold value; When the potential function is a radially symmetric function and its zero set is the unit ball in R3, and the nonlinearity is a power function, we provide a specific estimate value of the threshold value. At the same time, concentration behaviors of solutions are obtained. Then by the minimization argument on the sign-changing Nehari manifold and a quantitative deformation lemma, we prove that the system has a sign-changing solution. Moreover, concentration behaviors of sign-changing solutions are obtained when the coefficient of the potential function tends to infinity. Specially, our results of positive ground state solutions and sign-changing solutions cover general Schrodinger equations, Kir chhoff equations and Schrodinger-Poisson systems.Our system is the following form where a > 0,b, l ? 0 are constants, ? > 0 is a parameter, the potential function V ?C(R3, R+), where R+ =[0,?), ? ? (0, 3) and p ? [2, 3 + ?).In the second chapter of the paper, in order to obtain the existence and concentra-tion of system positive ground state solutions, we assume that V, f satisfying the following conditions:(V0) V ? C(R3,R+) and there is M0 > 0 such that the set ? := {x ? R3 : V(x) < M0}has finite Lebesgue measure, i.e. m(?) < ?;(V1)? := int V-1(0) is nonempty with a smooth boundary and ? = V-1(0);(f0) f ? C1(R+,R+) and there exist q ? (2,6) and C > 0 such that |f'(t)| ? C(1+|t|q-2)for all t ? R+;(f1) f(t)/t2p-1 is increasing on (0, ?) and limt?? f(t)/t2p-1 = ?.We get the following results.Theorem 1.2 Assume that (V0), (V1), (f0) and (f1) hold. Then there exists ?0 > 0 such that the system (KH) has a positive ground state solution u? when ? > ?0. Furthermore,u??u0 in H1(R3) as ? ? ? and u0 ? H01(?) is a positive ground state solution of the limit systemIn the third chapter, in order to obtain the existence and concentration of system sign-changing solutions, we assume that V, f satisfying the following conditions:(V0) V ? C(R3,R+) and for every M > 0, the set VM := {x ? R3 : V(x) ? M} has a finite Lebesgue measure, i.e. m(Vm) < ?;(V1)Q := int V-1(0) is nonempty with a smooth boundary and ? = V-1(0);(f0) f ? C1(R,R) and there exist q ? (2,6) and C > 0 such that |f'(t)|? C(1 + |t|q-2)for all t ? R;(f1) the function f(t)/(|t|2(p-1)t) is nonincreasing on (-?,0) and nondecreasing on(0, ?) respectively, limt?0 f(t)/(|t|2(p-1)t) = 0 and lim|t|??f(t)/(|t|2(p-1)t = ?.In this paper, the third chapter is published in SCI core journal ” Journal of Mathemat-ical Analysis and Applications". We get the following results.Theorem 1.8 Assume that (V0),(V1),(f0) and (f1) hold. Then the system (KH) has at least a sign-changing solution u? ? HV1(R3). Furthermore, for any sequence {?n} (?) (0,?)with ?n ? ?, then there exist a subsequence {u?n} and some u0 ? H01(?) such that u?n ? u0 in H1(R3) and u0 is a sign-changing solution of the limit system...