| In this paper,we mainly consider two problems.Firstly,we consider the following Schr¨odinger-Poisson system:{-△u + V(x)u + λ?(x)u = K(x)f(u),x ∈ R3,-△? = u2,x ∈ R3,where λ>0 is a parameter.We investigate the existence of ground state solutions and least energy sign-changing solutions of the Schr¨odinger-Poisson system in R3,and estimate the energy of the sign-changing solutions.Because the so-called nonlocal term λΦ(x)u is involving in the equation,the variational functional of the equation has totally different properties from the case of λ = 0.We prove the existence of ground state solutions by using Mountain Pass theorem,and combining constraint minimization on nodal Nehari manifold,we prove that the system possesses one least energy sign-changing solution uλ.Moreover,we regard λ as a parameter and give a convergence property of uλas λ ↘ 0.Secondly,we study the existence and multiplicity of solutions of the following Kirchhoff type problem:(a +∫R3(|?u|2+ V(x)u2)dx)[-△u + V(x)u] = K(x)|u|q-2u + f(x,u),x ∈ R3.where parameter a > 0,1 < q < 2.In critical point theorem,Clark theorem proves that coercive even functionals existence a list of negative critical values converging to 0.Zhaoli Liu and Zhiqiang Wang improve the theorem,concluding that the list of critical points of functional also converge to 0.We apply the improved Clark theorem to Kirchhoff type equation,and then we get its infinitely many solutions. |
